Introduction to Myself
Rong Luo
Associate Professor of Mathematics
Department of Mathematics
West Virginia University
Education
Ph.D. in Mathematics, West Virginia University, 2002
M. S. in Computer Science, West Virginia University, 2002
M. S. in Mathematics, University of Science and Technology of China,
1998
B. S. in Mathematics, University of Science and Technology of China, 1996
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Introduction to Myself
Experience
Associate Professor of Mathematics, West Virginia University, August
2012-present
Professor of Mathematics, Middle Tennessee State University, August
2011-July 2012
Associate Professor of Mathematics, Middle Tennessee State University,
August 2007-July 2011
Assistant Professor of Mathematics, Middle Tennessee State University,
August 2002-July 2007
Research Interests: Graph Theory, Combinatorics, Combinatorial Matrix
Theory, Applications of Graph Theory in Chemistry and Biology
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Edge Coloring of Graphs
Rong Luo
Department of Mathematics
West Virginia University
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Introduction
In mathematics and computer science, graph theory is the study of
graphs, which are mathematical structures used to model pairwise
relations between objects.
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Introduction
In mathematics and computer science, graph theory is the study of
graphs, which are mathematical structures used to model pairwise
relations between objects.
A graph in this context is made up of vertices or nodes and lines
called edges that connect them.
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Introduction
In mathematics and computer science, graph theory is the study of
graphs, which are mathematical structures used to model pairwise
relations between objects.
A graph in this context is made up of vertices or nodes and lines
called edges that connect them.
A graph may be undirected, meaning that there is no distinction
between the two vertices associated with each edge.
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Introduction
In mathematics and computer science, graph theory is the study of
graphs, which are mathematical structures used to model pairwise
relations between objects.
A graph in this context is made up of vertices or nodes and lines
called edges that connect them.
A graph may be undirected, meaning that there is no distinction
between the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
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Introduction
In mathematics and computer science, graph theory is the study of
graphs, which are mathematical structures used to model pairwise
relations between objects.
A graph in this context is made up of vertices or nodes and lines
called edges that connect them.
A graph may be undirected, meaning that there is no distinction
between the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
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Introduction
In mathematics and computer science, graph theory is the study of
graphs, which are mathematical structures used to model pairwise
relations between objects.
A graph in this context is made up of vertices or nodes and lines
called edges that connect them.
A graph may be undirected, meaning that there is no distinction
between the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges are
called “parallel edges”.
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Introduction
In mathematics and computer science, graph theory is the study of
graphs, which are mathematical structures used to model pairwise
relations between objects.
A graph in this context is made up of vertices or nodes and lines
called edges that connect them.
A graph may be undirected, meaning that there is no distinction
between the two vertices associated with each edge.
Directed graph: its edges may be directed from one vertex to another.
Graphs are one of the prime objects of study in discrete mathematics.
If there are two or more edges connecting two vertices, such edges are
called “parallel edges”.
A graph is simple if it has no parallel edges. Otherwise it is called a
multigraph or simply a graph.
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Example
v
Petersen graph
The degree of v is 3.
The maximum degree ∆ = 3.
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Applications in Computer Science
In computer science, graphs are used to represent networks of
communication, data organization, computational devices, the flow of
computation, etc.
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Applications in Computer Science
In computer science, graphs are used to represent networks of
communication, data organization, computational devices, the flow of
computation, etc.
One practical example: The link structure of a website could be
represented by a directed graph.
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Applications in Computer Science
In computer science, graphs are used to represent networks of
communication, data organization, computational devices, the flow of
computation, etc.
One practical example: The link structure of a website could be
represented by a directed graph.
The vertices are the web pages available at the website and a directed
edge from page A to page B exists if and only if A contains a link to
B.
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Applications in Computer Science
In computer science, graphs are used to represent networks of
communication, data organization, computational devices, the flow of
computation, etc.
One practical example: The link structure of a website could be
represented by a directed graph.
The vertices are the web pages available at the website and a directed
edge from page A to page B exists if and only if A contains a link to
B.
A similar approach can be taken to problems in travel, biology,
computer chip design, and many other fields.
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Applications in Computer Science
In computer science, graphs are used to represent networks of
communication, data organization, computational devices, the flow of
computation, etc.
One practical example: The link structure of a website could be
represented by a directed graph.
The vertices are the web pages available at the website and a directed
edge from page A to page B exists if and only if A contains a link to
B.
A similar approach can be taken to problems in travel, biology,
computer chip design, and many other fields.
The development of algorithms to handle graphs is therefore of major
interest in computer science. There, the transformation of graphs is
often formalized and represented by graph rewrite systems.
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Applications in Physics and Chemistry
Graph theory is also used to study molecules in chemistry and physics.
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Applications in Physics and Chemistry
Graph theory is also used to study molecules in chemistry and physics.
In chemistry a graph makes a natural model for a molecule, where
vertices represent atoms and edges bonds.
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Applications in Physics and Chemistry
Graph theory is also used to study molecules in chemistry and physics.
In chemistry a graph makes a natural model for a molecule, where
vertices represent atoms and edges bonds.
This approach is especially used in computer processing of molecular
structures, ranging from chemical editors to database searching.
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Applications in Physics and Chemistry
Graph theory is also used to study molecules in chemistry and physics.
In chemistry a graph makes a natural model for a molecule, where
vertices represent atoms and edges bonds.
This approach is especially used in computer processing of molecular
structures, ranging from chemical editors to database searching.
Chemical graph theory has been studied extensively.
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Applications in Social Science
Graph theory is also widely used in sociology as a way.
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Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many different
types of graphs:
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Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many different
types of graphs:
the Acquaintanceship and Friendship Graphs, these graphs are useful
for representing whether n people know each other.
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Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many different
types of graphs:
the Acquaintanceship and Friendship Graphs, these graphs are useful
for representing whether n people know each other.
The influence graph. This graph is used to model whether certain
people can influence the behavior of others.
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Applications in Social Science
Graph theory is also widely used in sociology as a way.
Under the umbrella of Social Network graphs there are many different
types of graphs:
the Acquaintanceship and Friendship Graphs, these graphs are useful
for representing whether n people know each other.
The influence graph. This graph is used to model whether certain
people can influence the behavior of others.
Collaboration graph which models whether two people work together
in a particular way.
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Graph Coloring
Graph Coloring is the most active area in graph theory.
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Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of a
graph (edges, vertices, faces) under certain rules to minimize the
number of color classes.
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Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of a
graph (edges, vertices, faces) under certain rules to minimize the
number of color classes.
It has many applications in practice.
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Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of a
graph (edges, vertices, faces) under certain rules to minimize the
number of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the vertices
of the graph in such a way that any two adjacent vertices receive
different colors.
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Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of a
graph (edges, vertices, faces) under certain rules to minimize the
number of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the vertices
of the graph in such a way that any two adjacent vertices receive
different colors.
Edge Coloring: An edge coloring of a graph is to color the edges of
the graph in such a way that any two adjacent edges receive different
colors.
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Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of a
graph (edges, vertices, faces) under certain rules to minimize the
number of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the vertices
of the graph in such a way that any two adjacent vertices receive
different colors.
Edge Coloring: An edge coloring of a graph is to color the edges of
the graph in such a way that any two adjacent edges receive different
colors.
Face coloring/Map Coloring
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Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of a
graph (edges, vertices, faces) under certain rules to minimize the
number of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the vertices
of the graph in such a way that any two adjacent vertices receive
different colors.
Edge Coloring: An edge coloring of a graph is to color the edges of
the graph in such a way that any two adjacent edges receive different
colors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimum
number of colors needed to color the objects.
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Graph Coloring
Graph Coloring is the most active area in graph theory.
In general, a graph coloring problem is to partition the objects of a
graph (edges, vertices, faces) under certain rules to minimize the
number of color classes.
It has many applications in practice.
Vertex coloring: A vertex coloring of a graph is to color the vertices
of the graph in such a way that any two adjacent vertices receive
different colors.
Edge Coloring: An edge coloring of a graph is to color the edges of
the graph in such a way that any two adjacent edges receive different
colors.
Face coloring/Map Coloring
The central problem in coloring problems is to find the minimum
number of colors needed to color the objects.
Graph Coloring has many applications in job scheduling, assignments
of classes/classrooms, assignments of wireless channels.
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Example
v
Petersen graph
The degree of v is 3.
The maximum degree ∆ = 3.
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Example-Vertex Coloring-3-cloring of the Peterson Graph
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Example–Edge Coloring
v
4−edge coloring of the petersen graph
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Example-Edge Coloring
v
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Example-Edge Coloring
v
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Example-Edge Coloring
v
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Example-Edge Coloring
v
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An Edge Coloring Application
Suppose transceivers u and w send a message to v . They must use
distinct frequencies. Otherwise, v will not be able to understand their
message as they will interfere with each other.
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An Edge Coloring Application
Suppose transceivers u and w send a message to v . They must use
distinct frequencies. Otherwise, v will not be able to understand their
message as they will interfere with each other.
Suppose that transceivers u wants to communicate with transceiver
v , transceiver w wants to communicate with transceiver x, and v and
w are close.
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An Edge Coloring Application
Suppose transceivers u and w send a message to v . They must use
distinct frequencies. Otherwise, v will not be able to understand their
message as they will interfere with each other.
Suppose that transceivers u wants to communicate with transceiver
v , transceiver w wants to communicate with transceiver x, and v and
w are close.
If u and w send messages on a same frequency, v will receive both
messages on the same frequency and so messages will interfere with
each color.
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An Edge Coloring Application
Suppose transceivers u and w send a message to v . They must use
distinct frequencies. Otherwise, v will not be able to understand their
message as they will interfere with each other.
Suppose that transceivers u wants to communicate with transceiver
v , transceiver w wants to communicate with transceiver x, and v and
w are close.
If u and w send messages on a same frequency, v will receive both
messages on the same frequency and so messages will interfere with
each color.
Solving the frequency assignment problem is equivalent to finding an
edge coloring satisfying: (1) adjacent edges are colored with different
colors and (2) two edges adjacent to a common edge are colored
differently.
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Edge Coloring of Simple Graphs
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Introduction
Tait first studied edge coloring problem in 1880 when he tried to
prove the Four Color Conjecture.
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Introduction
Tait first studied edge coloring problem in 1880 when he tried to
prove the Four Color Conjecture.
He proved that the Four Color Conjecture is equivalent to that every
2-edge connected planar cubic graph is edge 3-colorable.
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Four Color Theorem
Theorem
(Four Color Theorem, Appel and Haken, 1976) Every map can be colored
with four colors such that any two neighboring regions are colored with
different colors.
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Coloring the map of USA
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History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie in
October 1852 while trying to color the map of counties of England he
noticed that four colors sufficed.
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History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie in
October 1852 while trying to color the map of counties of England he
noticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can be
colored using four colors in such a way that adjacent regions receive
different colors.
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History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie in
October 1852 while trying to color the map of counties of England he
noticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can be
colored using four colors in such a way that adjacent regions receive
different colors.
Frederick Guthrie then communicated the conjecture to De Morgan.
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History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie in
October 1852 while trying to color the map of counties of England he
noticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can be
colored using four colors in such a way that adjacent regions receive
different colors.
Frederick Guthrie then communicated the conjecture to De Morgan.
The first printed reference is due to Cayley in 1878.
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History of the Four Color Theorem
The Four Colour Conjecture was first made by Francis Guthrie in
October 1852 while trying to color the map of counties of England he
noticed that four colors sufficed.
He asked his brother Frederick if it was true that any map can be
colored using four colors in such a way that adjacent regions receive
different colors.
Frederick Guthrie then communicated the conjecture to De Morgan.
The first printed reference is due to Cayley in 1878.
Appel and Haken proved the conjecture in 1976.
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Two fallacious proofs of the Four Color Theorem– Kempe
Kempe “proved” it in 1879.
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Two fallacious proofs of the Four Color Theorem– Kempe
Kempe “proved” it in 1879.
Found to be flawed by Heawood in 1890.
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Two fallacious proofs of the Four Color Theorem– Kempe
Kempe “proved” it in 1879.
Found to be flawed by Heawood in 1890.
Heawood proved that 5 colors are enough.
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Two fallacious proofs of the Four Color Theorem– Kempe
Kempe “proved” it in 1879.
Found to be flawed by Heawood in 1890.
Heawood proved that 5 colors are enough.
Kempe’s error proved very difficult to patch up, but in fact the
eventual solution used important ideas that can be traced back to
Kempe.
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Two fallacious proofs of the Four Color Theorem– Tait
Tait “proved” it in 1880.
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Two fallacious proofs of the Four Color Theorem– Tait
Tait “proved” it in 1880.
Found an equivalent formulation of the 4CT in terms of three-edge
coloring.
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Two fallacious proofs of the Four Color Theorem– Tait
Tait “proved” it in 1880.
Found an equivalent formulation of the 4CT in terms of three-edge
coloring.
Found to be flawed by Petersen in 1891.
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Two fallacious proofs of the Four Color Theorem– Tait
Tait “proved” it in 1880.
Found an equivalent formulation of the 4CT in terms of three-edge
coloring.
Found to be flawed by Petersen in 1891.
His results stimulated interest in edge-coloring.
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Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can be
colored with four colors.
• A cubic map is a map in which there are exactly three countries at each
meeting point.
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Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can be
colored with four colors.
• A cubic map is a map in which there are exactly three countries at each
meeting point.
Orginal Map
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Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can be
colored with four colors.
• A cubic map is a map in which there are exactly three countries at each
meeting point.
Orginal Map
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Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can be
colored with four colors.
• A cubic map is a map in which there are exactly three countries at each
meeting point.
A
B
D
A
B
C
Color the new map with 4 colors
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Cayley’s Reduction to Cubic Map
• Cayley observed that it is sufficient to prove that any cubic map can be
colored with four colors.
• A cubic map is a map in which there are exactly three countries at each
meeting point.
A
A
B
B
B
A
C
Remove the patch
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D
A
C
Color the new map with 4 colors
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Cayley’s Reduction to Cubic Map
Lemma
(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane map
is face 4-colorable.
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Tait’s Theorem
Lemma
(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane map
is face 4-colorable.
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Tait’s Theorem
Lemma
(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane map
is face 4-colorable.
Theorem
(Tait, 1880) Every cubic plane map is face 4-colorable ⇐⇒ Every cubic
plane map is edge 3-colorable.
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Tait’s Theorem
Lemma
(Cayley) Every plane map is face 4-colorable ⇐⇒ Every cubic plane map
is face 4-colorable.
Theorem
(Tait, 1880) Every cubic plane map is face 4-colorable ⇐⇒ Every cubic
plane map is edge 3-colorable.
Theorem
The four color theorem is equivalent to that every 2-edge connected cubic
plane graph is edge 3-colorable.
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Face 4-coloring and Edge 3-coloring
Theorem
(Tait, 1880) Every cubic plane map is face 4-colorable ⇐⇒ Every cubic
plane map is edge 3-colorable.
B
D
D
A
C
B
Edge 3−coloring
Face 4−coloring
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From face 4-coloring to edge 3-coloring
D
B
D
A
D
B
C
A
B
D
C
B
{AB, CD} = RED
Face 4−coloring to Edge 3−coloring
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From face 4-coloring to edge 3-coloring
D
B
D
A
D
B
C
A
B
D
C
B
{AB, CD} = RED
Face 4−coloring to Edge 3−coloring
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From face 4-coloring to edge 3-coloring
D
B
D
A
D
B
C
A
B
D
C
B
{AB, CD} = RED
{AC,BD} = BLUE
Face 4−coloring to Edge 3−coloring
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From face 4-coloring to edge 3-coloring
D
B
D
A
D
B
A
C
B
D
C
B
{AB, CD} = RED
{AC,BD} = BLUE
Face 4−coloring to Edge 3−coloring
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From face 4-coloring to edge 3-coloring
D
B
D
A
D
B
C
A
B
D
C
B
{AB, CD} = RED {AC,BD} = BLUE {AD,BC} = GREEN
Face 4−coloring to Edge 3−coloring
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From face 4-coloring to edge 3-coloring
D
B
D
A
D
B
C
A
B
D
C
B
{AB, CD} = RED {AC,BD} = BLUE {AD,BC} = GREEN
Face 4−coloring to Edge 3−coloring
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From face 4-coloring to edge 3-coloring
• The edges are colored with THREE colors: Red, Blue, Green
• Two edges sharing a common endvertex are colored with different
colors(?)
y
u
x
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From face 4-coloring to edge 3-coloring
• The edges are colored with THREE colors: Red, Blue, Green
• Two edges sharing a common endvertex are colored with different
colors(?)
y
u
x
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z
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From face 4-coloring to edge 3-coloring
• The edges are colored with THREE colors: Red, Blue, Green
• Two edges sharing a common endvertex are colored with different
colors(?)
y
A
B
u
C
x
{AB, CD} = RED
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z
{AC,BD} = BLUE
Edge Coloring of Graphs
{AD,BC} = GREEN
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From face 4-coloring to edge 3-coloring
• The edges are colored with THREE colors: Red, Blue, Green
• Two edges sharing a common endvertex are colored with different
colors(?)
y
A
B
u
C
x
{AB, CD} = RED
z
{AC,BD} = BLUE
{AD,BC} = GREEN
• If a cubic plane graph has a face 4-coloring, then it has an edge
3-coloring.
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From edge 3-coloring to face 4-coloring
D
B
D
D
A
B
C
B
A
D
C
B
{AB, CD} = RED
{AC, BD} = BLUE {AD, BC} = GREEN
Face 4−coloring to Edge 3−coloring
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From edge 3-coloring to face 4-coloring
D
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
Rong Luo (WVU)
Edge Coloring of Graphs
44 / 105
From edge 3-coloring to face 4-coloring
B
D
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
Rong Luo (WVU)
Edge Coloring of Graphs
45 / 105
From edge 3-coloring to face 4-coloring
D
B
D
B
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
Rong Luo (WVU)
Edge Coloring of Graphs
46 / 105
From edge 3-coloring to face 4-coloring
D
B
D
B
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
Rong Luo (WVU)
Edge Coloring of Graphs
47 / 105
From edge 3-coloring to face 4-coloring
D
B
D
A
C
B
{AB, CD} = RED {AC, BD} = BLUE {AD, BC} = GREEN
Edge 3−coloring to Face 4−coloring
Rong Luo (WVU)
Edge Coloring of Graphs
48 / 105
Introduction– Vizing’s Theorem
The basic question: Given a graph G , what is the smallest number of
colors needed to color the edges?
Rong Luo (WVU)
Edge Coloring of Graphs
49 / 105
Introduction– Vizing’s Theorem
The basic question: Given a graph G , what is the smallest number of
colors needed to color the edges?
This number, denoted χe (G ), is called the edge chromatic number of
G.
Rong Luo (WVU)
Edge Coloring of Graphs
49 / 105
A natural lower bound
Rong Luo (WVU)
Edge Coloring of Graphs
50 / 105
A natural lower bound
Rong Luo (WVU)
Edge Coloring of Graphs
51 / 105
A natural lower bound
At least ∆ colors are needed.
Rong Luo (WVU)
Edge Coloring of Graphs
52 / 105
A natural lower bound
At least ∆ colors are needed. So χe (G ) ≥ ∆. How big could χe (G ) be?
Rong Luo (WVU)
Edge Coloring of Graphs
53 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Rong Luo (WVU)
Edge Coloring of Graphs
54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe (G ) = ∆ or ∆ + 1.
Rong Luo (WVU)
Edge Coloring of Graphs
54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe (G ) = ∆ or ∆ + 1.
Classification of graphs:
Rong Luo (WVU)
Edge Coloring of Graphs
54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe (G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe (G ) = ∆
Rong Luo (WVU)
Edge Coloring of Graphs
54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe (G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe (G ) = ∆
Class two: χe (G ) = ∆ + 1
Rong Luo (WVU)
Edge Coloring of Graphs
54 / 105
Introduction– Vizing’s Theorem
In 1964, a breakthrough came.
Theorem
(Vizing’s Theorem) For each simple graph G , χe (G ) = ∆ or ∆ + 1.
Classification of graphs:
Class one: χe (G ) = ∆
Class two: χe (G ) = ∆ + 1
Theorem
(Holyer, 1980) The problem of determining whether a graph is class one or
class two is NP-hard.
Rong Luo (WVU)
Edge Coloring of Graphs
54 / 105
Examples
Rong Luo (WVU)
Edge Coloring of Graphs
55 / 105
Examples
Rong Luo (WVU)
Edge Coloring of Graphs
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Examples
Rong Luo (WVU)
Edge Coloring of Graphs
57 / 105
Examples
Rong Luo (WVU)
Edge Coloring of Graphs
58 / 105
Examples
Rong Luo (WVU)
Edge Coloring of Graphs
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Examples
An odd cycle is class two.
Rong Luo (WVU)
Edge Coloring of Graphs
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Examples
An even cycle is class one.
Rong Luo (WVU)
Edge Coloring of Graphs
61 / 105
Examples
An even cycle is class one.
Rong Luo (WVU)
Edge Coloring of Graphs
62 / 105
The Petersen graph is class two
v
4−edge coloring of the petersen graph
Rong Luo (WVU)
Edge Coloring of Graphs
63 / 105
The Petersen graph is class two
v
Petersen graph
Suppose that we can color the edges of the Petersen graph with three
colors: red, blue, and green.
Rong Luo (WVU)
Edge Coloring of Graphs
64 / 105
The Petersen graph is class two
The outer cycle sees all three colors.
v
The outer 5−cycle is colored with three colors
Rong Luo (WVU)
Edge Coloring of Graphs
65 / 105
The Petersen graph is class two
Each vertex sees all 3 colors.
v
Each vertex sees all three colors
Rong Luo (WVU)
Edge Coloring of Graphs
66 / 105
The Petersen graph is class two
v
Rong Luo (WVU)
Edge Coloring of Graphs
67 / 105
The Petersen graph is class two
v
Rong Luo (WVU)
Edge Coloring of Graphs
68 / 105
The Petersen graph is class two
v
X
u
w
X
Rong Luo (WVU)
Edge Coloring of Graphs
69 / 105
The Petersen graph is class two
v
u
w
Rong Luo (WVU)
Edge Coloring of Graphs
70 / 105
The Petersen graph is class two
v
u
w
Rong Luo (WVU)
Edge Coloring of Graphs
71 / 105
The Petersen graph is class two
v
u
w
Rong Luo (WVU)
Edge Coloring of Graphs
72 / 105
The Petersen graph is class two
v
u
w
Rong Luo (WVU)
Edge Coloring of Graphs
73 / 105
The Petersen graph is class two
v
u
w
There are at least two red edges on the inner cycle.
Rong Luo (WVU)
Edge Coloring of Graphs
74 / 105
The Petersen graph is class two
v
The outer 5−cycle is colored with three colors
There are at least two red edges, two blue edges, and two green edges
on the inner cycle.
Rong Luo (WVU)
Edge Coloring of Graphs
75 / 105
The Petersen graph is class two
v
The outer 5−cycle is colored with three colors
There are at least two red edges, two blue edges, and two green edges
on the inner cycle.
The inner cycle must have at least 6 edges.
Rong Luo (WVU)
Edge Coloring of Graphs
75 / 105
The Petersen graph is class two
v
The outer 5−cycle is colored with three colors
There are at least two red edges, two blue edges, and two green edges
on the inner cycle.
The inner cycle must have at least 6 edges.
The inner cycle is a 5-cycle and only has five edges!
Rong Luo (WVU)
Edge Coloring of Graphs
75 / 105
The Petersen graph is class two
v
The outer 5−cycle is colored with three colors
There are at least two red edges, two blue edges, and two green edges
on the inner cycle.
The inner cycle must have at least 6 edges.
The inner cycle is a 5-cycle and only has five edges!
This proves that the Petersen graph is not 3-edge colorable and thus
it is class two.
Rong Luo (WVU)
Edge Coloring of Graphs
75 / 105
Critical Graphs
G is critical (or ∆-critical) if χe (G ) = ∆ + 1 and χe (G − e) ≤ ∆ for
any edge e in G .
Rong Luo (WVU)
Edge Coloring of Graphs
76 / 105
Critical Graphs
G is critical (or ∆-critical) if χe (G ) = ∆ + 1 and χe (G − e) ≤ ∆ for
any edge e in G .
2-critical graphs are odd cycles.
Rong Luo (WVU)
Edge Coloring of Graphs
76 / 105
Critical Graphs
G is critical (or ∆-critical) if χe (G ) = ∆ + 1 and χe (G − e) ≤ ∆ for
any edge e in G .
2-critical graphs are odd cycles.
Criticality is a general concept in graph theory and can be defined
with respect to various graph parameters.
Rong Luo (WVU)
Edge Coloring of Graphs
76 / 105
Critical Graphs
G is critical (or ∆-critical) if χe (G ) = ∆ + 1 and χe (G − e) ≤ ∆ for
any edge e in G .
2-critical graphs are odd cycles.
Criticality is a general concept in graph theory and can be defined
with respect to various graph parameters.
The importance of the notion of criticality is that problems for graphs
in general may often be reduced to problems for critical graphs whose
structure is more restricted.
Rong Luo (WVU)
Edge Coloring of Graphs
76 / 105
Critical Graphs
G is critical (or ∆-critical) if χe (G ) = ∆ + 1 and χe (G − e) ≤ ∆ for
any edge e in G .
2-critical graphs are odd cycles.
Criticality is a general concept in graph theory and can be defined
with respect to various graph parameters.
The importance of the notion of criticality is that problems for graphs
in general may often be reduced to problems for critical graphs whose
structure is more restricted.
Critical graphs (with respect to the vertex chromatic number) were
first introduced and used by Dirac in 1951.
Rong Luo (WVU)
Edge Coloring of Graphs
76 / 105
Independence Number
An independent set is a set of vertices in which no pair of vertices are
connected with an edge. The size of a maximum independent set is called
independence number.
v
u
x
y
The red vertices u, v , x, y form an independent set.
The independence number of the Petersen graph is 4.
Rong Luo (WVU)
Edge Coloring of Graphs
77 / 105
Example
A 2-factor of a graph is a 2-regular spanning subgraph.
v
u
x
y
The blue cycles form a 2-factor.
Rong Luo (WVU)
Edge Coloring of Graphs
78 / 105
2-factor
A 2-factor of a graph is a 2-regular spanning subgraph.
The blue cycles form a 2-factor.
Rong Luo (WVU)
Edge Coloring of Graphs
79 / 105
Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all the vertices of
the graph.
Rong Luo (WVU)
Edge Coloring of Graphs
80 / 105
Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all the vertices of
the graph.
Rong Luo (WVU)
Edge Coloring of Graphs
81 / 105
Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all the
vertices of the graph.
Rong Luo (WVU)
Edge Coloring of Graphs
82 / 105
Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all the
vertices of the graph.
A hamiltonian cycle is a special 2-factor.
Rong Luo (WVU)
Edge Coloring of Graphs
82 / 105
Hamiltonian cycle
A hamiltonian cycle of a graph is a cycle that contains all the
vertices of the graph.
A hamiltonian cycle is a special 2-factor.
A graph is hamiltonian if it has a hamiltonian cycle.
Rong Luo (WVU)
Edge Coloring of Graphs
82 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures.
(Vizing’s Independence Number Conjecture) The independence
number of a critical graph is at most half of the number of vertices.
Rong Luo (WVU)
Edge Coloring of Graphs
83 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures.
(Vizing’s Independence Number Conjecture) The independence
number of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
Rong Luo (WVU)
Edge Coloring of Graphs
83 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures.
(Vizing’s Independence Number Conjecture) The independence
number of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The average
degree of a critical graph is at least ∆ − 1 + n3 .
Rong Luo (WVU)
Edge Coloring of Graphs
83 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures.
(Vizing’s Independence Number Conjecture) The independence
number of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The average
degree of a critical graph is at least ∆ − 1 + n3 .
(Vizing’s Planar Graph Conjecture) Every planar graph with
maximum degree 6 or 7 is class one.
Rong Luo (WVU)
Edge Coloring of Graphs
83 / 105
Edge coloring of Multigraphs
Rong Luo (WVU)
Edge Coloring of Graphs
84 / 105
History
Chapter 9, Twenty Pretty Edge Coloring Conjectures
Sample
Rong Luo (WVU)
Edge Coloring of Graphs
85 / 105
History
Chapter 9, Twenty Pretty Edge Coloring Conjectures
Sample
9 (late 1960s), 6 (1970s), 4 (early 1980s), 1 (1990)
Rong Luo (WVU)
Edge Coloring of Graphs
85 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which
is a generalization of Vizing Fan and Kiearstead path.
Rong Luo (WVU)
Edge Coloring of Graphs
86 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which
is a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Rong Luo (WVU)
Edge Coloring of Graphs
86 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which
is a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Kurt O. (2009) The Ohio State University, (Neil Robertson)
Rong Luo (WVU)
Edge Coloring of Graphs
86 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which
is a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Kurt O. (2009) The Ohio State University, (Neil Robertson)
MacDonald, J. (2009), University of Waterloo (Penny Haxell)
Rong Luo (WVU)
Edge Coloring of Graphs
86 / 105
History
Tashkinov, 2000, introduced a method, called Tashkinov tree, which
is a generalization of Vizing Fan and Kiearstead path.
Three Ph.D dissertations on edge coloring (Goldberg Conjecture)
Kurt O. (2009) The Ohio State University, (Neil Robertson)
MacDonald, J. (2009), University of Waterloo (Penny Haxell)
Scheide (2007), ( Michael Stiebitz)
Rong Luo (WVU)
Edge Coloring of Graphs
86 / 105
Edge chromatic number
χe (G ) ≥ ∆.
Rong Luo (WVU)
Edge Coloring of Graphs
87 / 105
Edge chromatic number
χe (G ) ≥ ∆.
How high can edge chromatic number be?
Rong Luo (WVU)
Edge Coloring of Graphs
87 / 105
Edge chromatic number
χe (G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe (G ) = ∆.
Rong Luo (WVU)
Edge Coloring of Graphs
87 / 105
Edge chromatic number
χe (G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe (G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Rong Luo (WVU)
Edge Coloring of Graphs
87 / 105
Edge chromatic number
χe (G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe (G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem)
For each simple graph G , χe (G ) = ∆ or ∆ + 1.
Rong Luo (WVU)
Edge Coloring of Graphs
87 / 105
Edge chromatic number
χe (G ) ≥ ∆.
How high can edge chromatic number be?
Theorem
(Konig, 1916) If G is a bipartite graph, then χe (G ) = ∆.
Konig’s result is the first result on edge coloring with a correct proof.
Theorem
(Vizing Theorem)
For each simple graph G , χe (G ) = ∆ or ∆ + 1.
For a simple graph G , χe (G ) ∈ {∆, ∆ + 1}.
Rong Luo (WVU)
Edge Coloring of Graphs
87 / 105
Edge chromatic number
Rong Luo (WVU)
Edge Coloring of Graphs
88 / 105
Edge chromatic number–Upper bounds
χe (G ) ≤
3∆
2
Rong Luo (WVU)
(Shannon, 1949).
Edge Coloring of Graphs
89 / 105
Edge chromatic number–Upper bounds
χe (G ) ≤
3∆
2
(Shannon, 1949).
χe (G ) ≤ ∆ + µ where µ is the multiplicity of G
Rong Luo (WVU)
Edge Coloring of Graphs
(Vizing, 1964).
89 / 105
Edge chromatic number–Upper bounds
χe (G ) ≤
3∆
2
(Shannon, 1949).
χe (G ) ≤ ∆ + µ where µ is the multiplicity of G
χe (G ) ≤ ∆ + d g∆−2
e
o −1
Rong Luo (WVU)
(Vizing, 1964).
(Goldberg, 1970s).
Edge Coloring of Graphs
89 / 105
Edge chromatic number–Upper bounds
χe (G ) ≤
3∆
2
(Shannon, 1949).
χe (G ) ≤ ∆ + µ where µ is the multiplicity of G
χe (G ) ≤ ∆ + d g∆−2
e
o −1
µ
χe (G ) ≤ ∆ + d b g c e
(Vizing, 1964).
(Goldberg, 1970s).
(Stephen, 2001).
2
Rong Luo (WVU)
Edge Coloring of Graphs
89 / 105
Edge chromatic number–Upper bounds
χe (G ) ≤
3∆
2
(Shannon, 1949).
χe (G ) ≤ ∆ + µ where µ is the multiplicity of G
χe (G ) ≤ ∆ + d g∆−2
e
o −1
µ
χe (G ) ≤ ∆ + d b g c e
(Vizing, 1964).
(Goldberg, 1970s).
(Stephen, 2001).
2
In general χe (G ) ∈ {∆, ∆ + 1, · · · , ∆ + µ}.
Rong Luo (WVU)
Edge Coloring of Graphs
89 / 105
Edge chromatic number–Upper bounds
χe (G ) ≤
3∆
2
(Shannon, 1949).
χe (G ) ≤ ∆ + µ where µ is the multiplicity of G
χe (G ) ≤ ∆ + d g∆−2
e
o −1
µ
χe (G ) ≤ ∆ + d b g c e
(Vizing, 1964).
(Goldberg, 1970s).
(Stephen, 2001).
2
In general χe (G ) ∈ {∆, ∆ + 1, · · · , ∆ + µ}.
For a simple graph G , χe (G ) ∈ {∆, ∆ + 1}.
Rong Luo (WVU)
Edge Coloring of Graphs
89 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1 , E2 , · · · Ek
where k = χe (G )
Rong Luo (WVU)
Edge Coloring of Graphs
90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1 , E2 , · · · Ek
where k = χe (G )
)|
Each color class is a matching, so |Ei | ≤ b |V (G
2 c.
Rong Luo (WVU)
Edge Coloring of Graphs
90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1 , E2 , · · · Ek
where k = χe (G )
)|
Each color class is a matching, so |Ei | ≤ b |V (G
2 c.
)|
The total number of edges in G , |E (G )| ≤ kb |V (G
2 c.
Rong Luo (WVU)
Edge Coloring of Graphs
90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1 , E2 , · · · Ek
where k = χe (G )
)|
Each color class is a matching, so |Ei | ≤ b |V (G
2 c.
)|
The total number of edges in G , |E (G )| ≤ kb |V (G
2 c.
χe (G ) = k ≥ d
Rong Luo (WVU)
|E (G )|
|V (G )| e.
b 2 c
Edge Coloring of Graphs
90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1 , E2 , · · · Ek
where k = χe (G )
)|
Each color class is a matching, so |Ei | ≤ b |V (G
2 c.
)|
The total number of edges in G , |E (G )| ≤ kb |V (G
2 c.
χe (G ) = k ≥ d
|E (G )|
|V (G )| e.
b 2 c
For any subgraph H of G , we have χe (G ) ≥ χe (H) ≥ d
Rong Luo (WVU)
Edge Coloring of Graphs
|E (H)|
|V (H)| e.
b 2 c
90 / 105
Edge chromatic number–Another nontrivial lower bound
Suppose G has an edge k-coloring with the color classes E1 , E2 , · · · Ek
where k = χe (G )
)|
Each color class is a matching, so |Ei | ≤ b |V (G
2 c.
)|
The total number of edges in G , |E (G )| ≤ kb |V (G
2 c.
χe (G ) = k ≥ d
|E (G )|
|V (G )| e.
b 2 c
For any subgraph H of G , we have χe (G ) ≥ χe (H) ≥ d
χe (G ) ≥
Rong Luo (WVU)
max
d
H⊆G ,|V (H)|≥2
Edge Coloring of Graphs
|E (H)|
b |V (H)|
2 c
|E (H)|
|V (H)| e.
b 2 c
e
90 / 105
Edge chromatic number–Another nontrivial lower bound
χe (G ) ≥
Rong Luo (WVU)
max
d
H⊆G ,|V (H)|≥2
Edge Coloring of Graphs
|E (H)|
b |V 2(H) |c
e
91 / 105
Edge chromatic number–Another nontrivial lower bound
χe (G ) ≥
w (G ) =
d
max
H⊆G ,|V (H)|≥2
max
d
H⊆G ,|V (H)|≥2
Rong Luo (WVU)
Edge Coloring of Graphs
|E (H)|
b |V 2(H) |c
|E (H)|
b |V (H)|
2 c
e
e
91 / 105
Edge chromatic number–Another nontrivial lower bound
χe (G ) ≥
w (G ) =
d
max
H⊆G ,|V (H)|≥2
max
d
H⊆G ,|V (H)|≥2
|E (H)|
b |V 2(H) |c
|E (H)|
b |V (H)|
2 c
e
e
w (G ) is called the density of G .
Rong Luo (WVU)
Edge Coloring of Graphs
91 / 105
Edge chromatic number–Another nontrivial lower bound
χe (G ) ≥
w (G ) =
d
max
H⊆G ,|V (H)|≥2
max
d
H⊆G ,|V (H)|≥2
|E (H)|
b |V 2(H) |c
|E (H)|
b |V (H)|
2 c
e
e
w (G ) is called the density of G .
χe (G ) ≥ w (G )
Rong Luo (WVU)
Edge Coloring of Graphs
91 / 105
Edge chromatic number–Another nontrivial lower bound
χe (G ) ≥
w (G ) =
d
max
H⊆G ,|V (H)|≥2
max
d
H⊆G ,|V (H)|≥2
|E (H)|
b |V 2(H) |c
|E (H)|
b |V (H)|
2 c
e
e
w (G ) is called the density of G .
χe (G ) ≥ w (G )
χe (G ) ≥ max{∆, w (G )}.
Rong Luo (WVU)
Edge Coloring of Graphs
91 / 105
Edge chromatic number–Another nontrivial lower bound
χe (G ) ≥
w (G ) =
d
max
H⊆G ,|V (H)|≥2
max
d
H⊆G ,|V (H)|≥2
|E (H)|
b |V 2(H) |c
|E (H)|
b |V (H)|
2 c
e
e
w (G ) is called the density of G .
χe (G ) ≥ w (G )
χe (G ) ≥ max{∆, w (G )}.
The maximum value can be achieved when |V (H)| is odd.
Rong Luo (WVU)
Edge Coloring of Graphs
91 / 105
Edge chromatic number–Another nontrivial lower bound
χe (G ) ≥
w (G ) =
d
max
H⊆G ,|V (H)|≥2
d
max
H⊆G ,|V (H)|≥2
|E (H)|
b |V 2(H) |c
|E (H)|
b |V (H)|
2 c
e
e
w (G ) is called the density of G .
χe (G ) ≥ w (G )
χe (G ) ≥ max{∆, w (G )}.
The maximum value can be achieved when |V (H)| is odd.
w (G ) =
max
d
|E (H)|
H⊆G ,|V (H)|≥2 b |V (H)| c
2
Rong Luo (WVU)
e=
max
H⊆G ,|V (H)|≥3,odd
Edge Coloring of Graphs
d
2|E (H)|
e
|V (H)| − 1
91 / 105
Seymour’s r -graph Conjecture
For simple graphs, ∆ ≤ χe (G ) ≤ ∆ + 1.
Rong Luo (WVU)
Edge Coloring of Graphs
92 / 105
Seymour’s r -graph Conjecture
For simple graphs, ∆ ≤ χe (G ) ≤ ∆ + 1.
In general, χe (G ) ≥ max{∆, w (G )}.
Rong Luo (WVU)
Edge Coloring of Graphs
92 / 105
Seymour’s r -graph Conjecture
For simple graphs, ∆ ≤ χe (G ) ≤ ∆ + 1.
In general, χe (G ) ≥ max{∆, w (G )}.
Is it true χe (G ) ≤ max{∆, w (G )} + 1.?
Rong Luo (WVU)
Edge Coloring of Graphs
92 / 105
Seymour’s r -graph Conjecture
For simple graphs, ∆ ≤ χe (G ) ≤ ∆ + 1.
In general, χe (G ) ≥ max{∆, w (G )}.
Is it true χe (G ) ≤ max{∆, w (G )} + 1.?
Conjecture
(Seymour’s r -graph Conjecture, 1979) Let G be a graph. Then
χe (G ) ≤ max{∆, w (G )} + 1.
Rong Luo (WVU)
Edge Coloring of Graphs
92 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe (G ) ≤ max{∆ + 1, w (G )}.
Rong Luo (WVU)
Edge Coloring of Graphs
93 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe (G ) ≤ max{∆ + 1, w (G )}.
Goldberg conjecture was proposed by Goldberg in 1970 and
independently by Seymour in 1979.
Rong Luo (WVU)
Edge Coloring of Graphs
93 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe (G ) ≤ max{∆ + 1, w (G )}.
Goldberg conjecture was proposed by Goldberg in 1970 and
independently by Seymour in 1979.
Goldberg Conjecture is equivalent to the following statements.
Rong Luo (WVU)
Edge Coloring of Graphs
93 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe (G ) ≤ max{∆ + 1, w (G )}.
Goldberg conjecture was proposed by Goldberg in 1970 and
independently by Seymour in 1979.
Goldberg Conjecture is equivalent to the following statements.
Let G be a graph. If χe (G ) ≥ ∆ + 2, then χe (G ) = w (G ).
Rong Luo (WVU)
Edge Coloring of Graphs
93 / 105
Goldberg Conjecture
Conjecture
(Goldberg Conjecture) Let G be a graph. Then
χe (G ) ≤ max{∆ + 1, w (G )}.
Goldberg conjecture was proposed by Goldberg in 1970 and
independently by Seymour in 1979.
Goldberg Conjecture is equivalent to the following statements.
Let G be a graph. If χe (G ) ≥ ∆ + 2, then χe (G ) = w (G ).
χe (G ) ∈ {∆, ∆ + 1, w (G )}.
Rong Luo (WVU)
Edge Coloring of Graphs
93 / 105
Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe (G ) ≥ ∆ + 2, then there is a
polynomial algorithm to compute χe (G ).
Rong Luo (WVU)
Edge Coloring of Graphs
94 / 105
Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe (G ) ≥ ∆ + 2, then there is a
polynomial algorithm to compute χe (G ).
So it implies that the difficulty in determining χe (G ) is only to
distinguish between two cases χe (G ) = ∆ and χe (G ) = ∆ + 1, which
is NP-hard proved by Holyer in 1980.
Rong Luo (WVU)
Edge Coloring of Graphs
94 / 105
Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe (G ) ≥ ∆ + 2, then there is a
polynomial algorithm to compute χe (G ).
So it implies that the difficulty in determining χe (G ) is only to
distinguish between two cases χe (G ) = ∆ and χe (G ) = ∆ + 1, which
is NP-hard proved by Holyer in 1980.
Goldberg Conjecture also implies Seymour’s r -graph conjecture and
Jakobsen’s critical graph conjecture.
Rong Luo (WVU)
Edge Coloring of Graphs
94 / 105
Goldberg Conjecture–Importance
Goldberg Conjecture implies that if χe (G ) ≥ ∆ + 2, then there is a
polynomial algorithm to compute χe (G ).
So it implies that the difficulty in determining χe (G ) is only to
distinguish between two cases χe (G ) = ∆ and χe (G ) = ∆ + 1, which
is NP-hard proved by Holyer in 1980.
Goldberg Conjecture also implies Seymour’s r -graph conjecture and
Jakobsen’s critical graph conjecture.
G is critical if χe (G − e) < χe (G ) for any edge e.
Rong Luo (WVU)
Edge Coloring of Graphs
94 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |
odd, |∂G (X )| ≥ r .
Rong Luo (WVU)
Edge Coloring of Graphs
95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |
odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfies
χe (G ) ≤ r + 1.
Rong Luo (WVU)
Edge Coloring of Graphs
95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |
odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfies
χe (G ) ≤ r + 1.
w (G ) ≤ r for each r -graph.
Rong Luo (WVU)
Edge Coloring of Graphs
95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |
odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfies
χe (G ) ≤ r + 1.
w (G ) ≤ r for each r -graph.
Let H ⊆ G with |V (H)| odd. Let X = V (H). Then
2|E (H)| ≤ r |X | − r = r (|X | − 1)
Rong Luo (WVU)
Edge Coloring of Graphs
95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |
odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfies
χe (G ) ≤ r + 1.
w (G ) ≤ r for each r -graph.
Let H ⊆ G with |V (H)| odd. Let X = V (H). Then
2|E (H)| ≤ r |X | − r = r (|X | − 1)
2|E (H)|
r (|X |−1)
|V (H)|−1 ≤ |X |−1 = r .
Rong Luo (WVU)
Edge Coloring of Graphs
95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |
odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfies
χe (G ) ≤ r + 1.
w (G ) ≤ r for each r -graph.
Let H ⊆ G with |V (H)| odd. Let X = V (H). Then
2|E (H)| ≤ r |X | − r = r (|X | − 1)
2|E (H)|
r (|X |−1)
|V (H)|−1 ≤ |X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be an
r -graph. (Why?)
Rong Luo (WVU)
Edge Coloring of Graphs
95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |
odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfies
χe (G ) ≤ r + 1.
w (G ) ≤ r for each r -graph.
Let H ⊆ G with |V (H)| odd. Let X = V (H). Then
2|E (H)| ≤ r |X | − r = r (|X | − 1)
2|E (H)|
r (|X |−1)
|V (H)|−1 ≤ |X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be an
r -graph. (Why?)
Seymour proved that every graph with max{∆, w (G )} ≤ r is
contained in an r -graph as a subgraph.
Rong Luo (WVU)
Edge Coloring of Graphs
95 / 105
Seymour’s r -graph Conjecture–Original Version
A r -regular graph is an r -graph if for every set X ⊆ V (G ) with |X |
odd, |∂G (X )| ≥ r .
Conjecture
(Seymour’s r -graph Conjecture, 1979) Every r -graph satisfies
χe (G ) ≤ r + 1.
w (G ) ≤ r for each r -graph.
Let H ⊆ G with |V (H)| odd. Let X = V (H). Then
2|E (H)| ≤ r |X | − r = r (|X | − 1)
2|E (H)|
r (|X |−1)
|V (H)|−1 ≤ |X |−1 = r .
If an r -regular graph has an edge r -coloring, then it must be an
r -graph. (Why?)
Seymour proved that every graph with max{∆, w (G )} ≤ r is
contained in an r -graph as a subgraph.
Since w (G ) ≤ r for each r -graph, Goldberg Conjecture implies
χe (G ) ≤ max{∆ + 1, w (G )} = r + 1.
Rong Luo (WVU)
Edge Coloring of Graphs
95 / 105
Seymour’s r -graph Conjecture–Equivalent Version
Seymour’s r -graph conjecture is equivalent to
Rong Luo (WVU)
Edge Coloring of Graphs
96 / 105
Seymour’s r -graph Conjecture–Equivalent Version
Seymour’s r -graph conjecture is equivalent to
Every graph G satisfies χe (G ) ≤ max{∆, w (G )} + 1.
Rong Luo (WVU)
Edge Coloring of Graphs
96 / 105
Seymour’s r -graph Conjecture–Equivalent Version
Seymour’s r -graph conjecture is equivalent to
Every graph G satisfies χe (G ) ≤ max{∆, w (G )} + 1.
Seymour’s r -graph conjecture is true for r ≤ 15.
Rong Luo (WVU)
Edge Coloring of Graphs
96 / 105
Seymour’s r -graph Conjecture–Equivalent Version
Seymour’s r -graph conjecture is equivalent to
Every graph G satisfies χe (G ) ≤ max{∆, w (G )} + 1.
Seymour’s r -graph conjecture is true for r ≤ 15.
Seymour’s r -graph conjecture suggests that if G is an r -graph then,
for all t ≥ 1, either χe (tG ) = max{∆(tG ), w (tG )} or
χe (tG ) = max{∆(tG ), w (tG )} + 1.
Rong Luo (WVU)
Edge Coloring of Graphs
96 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
The cases r = 0, 1, 2 are trivial.
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in
1990.
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in
1990.
Marcotte verified the conjecture for the class of graphs not containing
K3,3 or K5− as a minor, 2001.
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in
1990.
Marcotte verified the conjecture for the class of graphs not containing
K3,3 or K5− as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in
1990.
Marcotte verified the conjecture for the class of graphs not containing
K3,3 or K5− as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in
1990.
Marcotte verified the conjecture for the class of graphs not containing
K3,3 or K5− as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in
1990.
Marcotte verified the conjecture for the class of graphs not containing
K3,3 or K5− as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Seymour’s Exact Conjecture
For planar graph, Seymour proposed a stronger conjecture.
Conjecture
(Seymour’s Exact Conjecture, 1979) Every planar graph G satisfies
χe (G ) = max{∆, w (G )}.
The cases r = 0, 1, 2 are trivial.
Seymour proved the exact conjecture for series-parrallel graphs in
1990.
Marcotte verified the conjecture for the class of graphs not containing
K3,3 or K5− as a minor, 2001.
The case r = 3 is equivalent to the Four Color Theorem.
The cases r = 4, 5 was proved by Guenin (2011).
Dvorak, Kawarabayashi, and Kral solved the case r = 6 in 2011
The case of k = 7 was proved by Edwards and Kawarabayashi in 2012.
Chudnovsky, Edwards and Seymour solved the case k = 8 in 2012.
Rong Luo (WVU)
Edge Coloring of Graphs
97 / 105
Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph,
and let
m
m−3
χe (G ) >
∆(G ) +
.
m−1
m−1
for an odd integer m ≥ 3. Then |V (G )| ≤ m − 2.
Rong Luo (WVU)
Edge Coloring of Graphs
98 / 105
Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph,
and let
m
m−3
χe (G ) >
∆(G ) +
.
m−1
m−1
for an odd integer m ≥ 3. Then |V (G )| ≤ m − 2.
Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound.
Rong Luo (WVU)
Edge Coloring of Graphs
98 / 105
Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph,
and let
m
m−3
χe (G ) >
∆(G ) +
.
m−1
m−1
for an odd integer m ≥ 3. Then |V (G )| ≤ m − 2.
Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound.
Anderson proved that Goldberg’s Conjecture implies Jakobsen’s
conjecture.
Rong Luo (WVU)
Edge Coloring of Graphs
98 / 105
Jakobsen’s critical graph conjecture
Conjecture
(Jakobosen’s Critical Graph Conjecture, 1973) Let G be a critical graph,
and let
m
m−3
χe (G ) >
∆(G ) +
.
m−1
m−1
for an odd integer m ≥ 3. Then |V (G )| ≤ m − 2.
Jakobsen’s conjecture is trivial for m = 3 by Shannons upper bound.
Anderson proved that Goldberg’s Conjecture implies Jakobsen’s
conjecture.
Jakobsen’s conjecture was verified for 5 ≤ m ≤ 15.
Rong Luo (WVU)
Edge Coloring of Graphs
98 / 105
Fractional edge chromatic number
An edge coloring can be considered as an Integer Programming
Problem.
Rong Luo (WVU)
Edge Coloring of Graphs
99 / 105
Fractional edge chromatic number
An edge coloring can be considered as an Integer Programming
Problem.
Let M denote the set of all matchings of a graph G .
Rong Luo (WVU)
Edge Coloring of Graphs
99 / 105
Fractional edge chromatic number
An edge coloring can be considered as an Integer Programming
Problem.
Let M denote the set of all matchings of a graph G .
For each edge e, let Me denote the set of all matchings containing e.
Rong Luo (WVU)
Edge Coloring of Graphs
99 / 105
Fractional edge chromatic number
An edge coloring can be considered as an Integer Programming
Problem.
Let M denote the set of all matchings of a graph G .
For each edge e, let Me denote the set of all matchings containing e.
χe (G ) = min
X
yM ,
M∈M
subject
P to:
(1) M∈Me yM = 1 for each edge e ∈ E (G ).
(2) yM ∈ {0, 1}
Rong Luo (WVU)
Edge Coloring of Graphs
99 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e (G ) is defined as:
Rong Luo (WVU)
Edge Coloring of Graphs
100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e (G ) is defined as:
χ∗e (G ) = min
X
yM ,
M∈M
subject
P to:
(1) M∈Me yM = 1 for each edge e ∈ E (G ).
(2) 0 ≤ yM ≤ 1
Rong Luo (WVU)
Edge Coloring of Graphs
100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e (G ) is defined as:
χ∗e (G ) = min
X
yM ,
M∈M
subject
P to:
(1) M∈Me yM = 1 for each edge e ∈ E (G ).
(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edge
chromatic number in polynomial time.
Rong Luo (WVU)
Edge Coloring of Graphs
100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e (G ) is defined as:
χ∗e (G ) = min
X
yM ,
M∈M
subject
P to:
(1) M∈Me yM = 1 for each edge e ∈ E (G ).
(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edge
chromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e (G ) = max{∆,
Rong Luo (WVU)
max
|E (H)|
H⊆G ,|V (H)|≥2 b 1 |V (H)|c
2
Edge Coloring of Graphs
}.
100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e (G ) is defined as:
χ∗e (G ) = min
X
yM ,
M∈M
subject
P to:
(1) M∈Me yM = 1 for each edge e ∈ E (G ).
(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edge
chromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e (G ) = max{∆,
max
|E (H)|
H⊆G ,|V (H)|≥2 b 1 |V (H)|c
2
}.
If χ∗e (G ) = ∆, then w (G ) ≤ ∆
Rong Luo (WVU)
Edge Coloring of Graphs
100 / 105
Fractional chromatic index
The fractional edge chromatic number χ∗e (G ) is defined as:
χ∗e (G ) = min
X
yM ,
M∈M
subject
P to:
(1) M∈Me yM = 1 for each edge e ∈ E (G ).
(2) 0 ≤ yM ≤ 1
With matching techniques one can compute the fractional edge
chromatic number in polynomial time.
From Edmond’s matching polytope theorem,
χ∗e (G ) = max{∆,
max
|E (H)|
H⊆G ,|V (H)|≥2 b 1 |V (H)|c
2
}.
If χ∗e (G ) = ∆, then w (G ) ≤ ∆
If χ∗e (G ) > ∆, then w (G ) = dχ∗e (G )e and w (G ) can be computed in
polynomial time.
Rong Luo (WVU)
Edge Coloring of Graphs
100 / 105
Fractional chromatic index
The computation of the edge chromatic number χe (G ) is NP-hard
Rong Luo (WVU)
Edge Coloring of Graphs
101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe (G ) is NP-hard
The fractional edge chromatic number can be computed in
polynomial time.
Rong Luo (WVU)
Edge Coloring of Graphs
101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe (G ) is NP-hard
The fractional edge chromatic number can be computed in
polynomial time.
It is not clear whether the density w (G ) can be computed in
polynomial time.
Rong Luo (WVU)
Edge Coloring of Graphs
101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe (G ) is NP-hard
The fractional edge chromatic number can be computed in
polynomial time.
It is not clear whether the density w (G ) can be computed in
polynomial time.
max{∆(G ), w (G )} can be computed in polynomial time
Rong Luo (WVU)
Edge Coloring of Graphs
101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe (G ) is NP-hard
The fractional edge chromatic number can be computed in
polynomial time.
It is not clear whether the density w (G ) can be computed in
polynomial time.
max{∆(G ), w (G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe (G ) = dχ∗e (G )e
for every graph G with χe (G ) ≥ ∆ + 2.
Rong Luo (WVU)
Edge Coloring of Graphs
101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe (G ) is NP-hard
The fractional edge chromatic number can be computed in
polynomial time.
It is not clear whether the density w (G ) can be computed in
polynomial time.
max{∆(G ), w (G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe (G ) = dχ∗e (G )e
for every graph G with χe (G ) ≥ ∆ + 2.
p
χe (G ) ≤ χ∗e (G ) + χ∗e (G )/2 (Schiede, Sanders and Steuer 9/2 ).
Rong Luo (WVU)
Edge Coloring of Graphs
101 / 105
Fractional chromatic index
The computation of the edge chromatic number χe (G ) is NP-hard
The fractional edge chromatic number can be computed in
polynomial time.
It is not clear whether the density w (G ) can be computed in
polynomial time.
max{∆(G ), w (G )} can be computed in polynomial time
Goldberg Conjecture is equivalent to the claim that χe (G ) = dχ∗e (G )e
for every graph G with χe (G ) ≥ ∆ + 2.
p
χe (G ) ≤ χ∗e (G ) + χ∗e (G )/2 (Schiede, Sanders and Steuer 9/2 ).
Kahn proved in 1996 that every graph G satisfies χe (G ) ∼ χ∗e (G ) as
χ∗e (G ) → ∞.
Rong Luo (WVU)
Edge Coloring of Graphs
101 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe (G ) > 54 ∆ + 24 .
Rong Luo (WVU)
Edge Coloring of Graphs
102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe (G ) > 54 ∆ + 24 .
( Anderson, 1977) χe (G ) > 76 ∆ + 46 .
Rong Luo (WVU)
Edge Coloring of Graphs
102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe (G ) > 54 ∆ + 24 .
( Anderson, 1977) χe (G ) > 76 ∆ + 46 .
(Goldberg, 1984) χe (G ) > 98 ∆ + 86 .
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Edge Coloring of Graphs
102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe (G ) > 54 ∆ + 24 .
( Anderson, 1977) χe (G ) > 76 ∆ + 46 .
(Goldberg, 1984) χe (G ) > 98 ∆ + 86 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe (G ) >
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Edge Coloring of Graphs
11
10 ∆
+
8
10 .
102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe (G ) > 54 ∆ + 24 .
( Anderson, 1977) χe (G ) > 76 ∆ + 46 .
(Goldberg, 1984) χe (G ) > 98 ∆ + 86 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe (G ) >
(Favrholdt, Stiebitz, Toft, 2006) χe (G ) >
Rong Luo (WVU)
Edge Coloring of Graphs
13
12 ∆
+
11
10 ∆
+
8
10 .
10
12 .
102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe (G ) > 54 ∆ + 24 .
( Anderson, 1977) χe (G ) > 76 ∆ + 46 .
(Goldberg, 1984) χe (G ) > 98 ∆ + 86 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe (G ) >
(Favrholdt, Stiebitz, Toft, 2006) χe (G ) >
(Scheide, Stiebitz, 2009) χe (G ) >
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15
14 ∆
+
Edge Coloring of Graphs
13
12 ∆
12
14 .
+
11
10 ∆
+
8
10 .
10
12 .
102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe (G ) > 54 ∆ + 24 .
( Anderson, 1977) χe (G ) > 76 ∆ + 46 .
(Goldberg, 1984) χe (G ) > 98 ∆ + 86 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe (G ) >
(Favrholdt, Stiebitz, Toft, 2006) χe (G ) >
(Scheide, Stiebitz, 2009) χe (G ) >
15
14 ∆
+
13
12 ∆
12
14 .
+
Edge Coloring of Graphs
+
8
10 .
10
12 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe (G ) > ∆(G ) +
Rong Luo (WVU)
11
10 ∆
q
∆
2.
102 / 105
Progress toward Goldberg’s Conjecture
Goldberg’s conjecture has been verified for the following cases:
(Goldberg, 1974; Anderson, 1977) χe (G ) > 54 ∆ + 24 .
( Anderson, 1977) χe (G ) > 76 ∆ + 46 .
(Goldberg, 1984) χe (G ) > 98 ∆ + 86 .
(Nishizeki, Kashiwagi, 1990; Tashkinov, 2000) χe (G ) >
(Favrholdt, Stiebitz, Toft, 2006) χe (G ) >
(Scheide, Stiebitz, 2009) χe (G ) >
15
14 ∆
+
13
12 ∆
12
14 .
+
Edge Coloring of Graphs
+
8
10 .
10
12 .
(Chen, Yu, Zang, 2011; Schied, 2010) χe (G ) > ∆(G ) +
q
(Kurt, 2009) χe (G ) > ∆(G ) + 3 ∆
2.
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11
10 ∆
q
∆
2.
102 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLE
graphs.
(Vizing’s Independence Number Conjecture) The independence
number of a critical graph is at most half of the number of vertices.
Rong Luo (WVU)
Edge Coloring of Graphs
103 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLE
graphs.
(Vizing’s Independence Number Conjecture) The independence
number of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
Rong Luo (WVU)
Edge Coloring of Graphs
103 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLE
graphs.
(Vizing’s Independence Number Conjecture) The independence
number of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The average
degree of a critical graph is at least ∆ − 1 + n3 .
Rong Luo (WVU)
Edge Coloring of Graphs
103 / 105
Vizing’s Four Conjectures
In later 1960s, Vizing proposed the following four conjectures for SIMPLE
graphs.
(Vizing’s Independence Number Conjecture) The independence
number of a critical graph is at most half of the number of vertices.
(Vizing’s 2-factor Conjecture) Every critical graph has a 2-factor.
(Vizing’s Conjecture on the Size of Critical Graphs) The average
degree of a critical graph is at least ∆ − 1 + n3 .
(Vizing’s Planar Graph Conjecture) Every planar graph with
maximum degree 6 or 7 is class one.
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Edge Coloring of Graphs
103 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G
contains a family of six perfect matchings such that each edge of G is
contained in precisely two of the matchings.
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Edge Coloring of Graphs
104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G
contains a family of six perfect matchings such that each edge of G is
contained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe (2G ) = 6.
Rong Luo (WVU)
Edge Coloring of Graphs
104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G
contains a family of six perfect matchings such that each edge of G is
contained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe (2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a family
of five perfect matchings such that each edge of G is contained in at
least one of the matchings.
Rong Luo (WVU)
Edge Coloring of Graphs
104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G
contains a family of six perfect matchings such that each edge of G is
contained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe (2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a family
of five perfect matchings such that each edge of G is contained in at
least one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to The
Berge-Fulkerson Conjecture.
Rong Luo (WVU)
Edge Coloring of Graphs
104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G
contains a family of six perfect matchings such that each edge of G is
contained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe (2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a family
of five perfect matchings such that each edge of G is contained in at
least one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to The
Berge-Fulkerson Conjecture.
Matching cover:
∪k M |
mk (G ) = max{ |Ei=1(G )|i |M1 , M2 , · · · , Mk are perfect matchings ofG }.
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Edge Coloring of Graphs
104 / 105
Other Conjectures
The Berge-Fulkerson Conjecture Every bridge less cubic graph G
contains a family of six perfect matchings such that each edge of G is
contained in precisely two of the matchings.
(Equivalence) For each bridge less cubic graph G , χe (2G ) = 6.
(Berge’s conjecture) Every bridgeless cubic graph G contains a family
of five perfect matchings such that each edge of G is contained in at
least one of the matchings.
Mazzuoccolo proved that Berge’s conjecture is equivalent to The
Berge-Fulkerson Conjecture.
Matching cover:
∪k M |
mk (G ) = max{ |Ei=1(G )|i |M1 , M2 , · · · , Mk are perfect matchings ofG }.
m1 (G ) =
1
3
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and Berge’s conjecture suggests that m5 (G ) = 1.
Edge Coloring of Graphs
104 / 105
Other conjectures see Chapter 9 of the book: Twenty Pretty Edge
Coloring Conjectures
Sample
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Edge Coloring of Graphs
105 / 105