Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Reed’s Conjecture
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Naveen Sundar G.
April 15, 2010
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
The Outline
Reed’s
Conjecture
1
Preliminaries
Basic Results
The Conjecture
General Properties
2
Line Graphs
Definitions and Observations
Fractional vertex c-coloring
List Coloring
Chromatic Index and Fractional Chromatic Index
The Theorem and Proof
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Case I
Case II
3
Similar Conjectures
Naveen Sundar G.
Reed’s Conjecture
ω, ∆ & χ
Reed’s
Conjecture
Naveen
Sundar G.
Definitions
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
ω, ∆ & χ
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Definitions
1 The clique number ω(G ) of a graph G is the size of the
largest clique in that graph.
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
ω, ∆ & χ
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Definitions
1 The clique number ω(G ) of a graph G is the size of the
largest clique in that graph.
Preliminaries
Basic Results
The Conjecture
General
Properties
2
The maximum degree ∆(G ) of a graph G is the
maximum number of neighbors of any vertex in that graph.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
ω, ∆ & χ
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Definitions
1 The clique number ω(G ) of a graph G is the size of the
largest clique in that graph.
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
2
The maximum degree ∆(G ) of a graph G is the
maximum number of neighbors of any vertex in that graph.
3
The chromatic number χ(G ) of a graph G is the
minimum number of colors needed to color the vertices in
that graph so that adjacent vertices have different colors.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
ω, ∆ & χ
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Definitions
1 The clique number ω(G ) of a graph G is the size of the
largest clique in that graph.
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
2
The maximum degree ∆(G ) of a graph G is the
maximum number of neighbors of any vertex in that graph.
3
The chromatic number χ(G ) of a graph G is the
minimum number of colors needed to color the vertices in
that graph so that adjacent vertices have different colors.
4
The chromatic index χe (G ) of a graph G is the
minimum number of colors needed to color the edges in
that graph so that adjacent edges have different colors.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Some basic bounds
Reed’s
Conjecture
Naveen
Sundar G.
The presence of a clique sets a lower bound on the maximum
degree
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Bound
Upper Bound
Lower Bound
Line Graphs
∆
None
ω−1
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
ω
∆+1
None
Some basic bounds
Reed’s
Conjecture
Naveen
Sundar G.
The presence of a clique sets a lower bound on the maximum
degree
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Bound
Upper Bound
Lower Bound
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
∆
None
ω−1
We also have ω(G ) ≤ χ(G ) ≤ ∆(G ) + 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
ω
∆+1
None
Some basic bounds
Reed’s
Conjecture
Naveen
Sundar G.
The presence of a clique sets a lower bound on the maximum
degree
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Bound
Upper Bound
Lower Bound
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
∆
None
ω−1
ω
∆+1
None
We also have ω(G ) ≤ χ(G ) ≤ ∆(G ) + 1
Theorem (Brooks)
For graphs other than cliques and with ∆(G ) ≥ 3 we have
χ(G ) ≤ ∆(G )
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Reed’s Conjecture , 1998
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
The Conjecture :
χ(G ) ≤ d 1+∆+ω
e
2
The conjecture holds for ω ∈ {2, ∆ − 1, ∆, ∆ + 1}
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Some Observations
Reed’s
Conjecture
Naveen
Sundar G.
1
More densely connected a graph is the more colors we
have to use to color the graph. Complete subgraphs
increase the number of colors.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Some Observations
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
Preliminaries
2
More densely connected a graph is the more colors we
have to use to color the graph. Complete subgraphs
increase the number of colors.
The only cases where we need ∆ + 1 coloring are when the
graph is a clique or an odd cycle.
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Some Observations
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
Preliminaries
Basic Results
The Conjecture
General
Properties
3
More densely connected a graph is the more colors we
have to use to color the graph. Complete subgraphs
increase the number of colors.
The only cases where we need ∆ + 1 coloring are when the
graph is a clique or an odd cycle.
The clique number can be very small and the maximum
degree can grow pretty large.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Some Observations
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
Preliminaries
Basic Results
The Conjecture
General
Properties
3
More densely connected a graph is the more colors we
have to use to color the graph. Complete subgraphs
increase the number of colors.
The only cases where we need ∆ + 1 coloring are when the
graph is a clique or an odd cycle.
The clique number can be very small and the maximum
degree can grow pretty large.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Some Observations
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
Preliminaries
Basic Results
The Conjecture
General
Properties
3
More densely connected a graph is the more colors we
have to use to color the graph. Complete subgraphs
increase the number of colors.
The only cases where we need ∆ + 1 coloring are when the
graph is a clique or an odd cycle.
The clique number can be very small and the maximum
degree can grow pretty large.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Figure: Omega and Delta
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
The Conjecture For Line Graphs
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
An upper bound for the chromatic number of line graphs
by
A.D. King, B.A. Reed, A. Vetta
2006.
http://www.columbia.edu/~ak3074/papers/
KingReedVetta-linegraphs.ps
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definitions and Observations
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definitions and Observations
Reed’s
Conjecture
Naveen
Sundar G.
Multigraph A multigraph H is a graph in which there can be
multiple edges between two nodes.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definitions and Observations
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Multigraph A multigraph H is a graph in which there can be
multiple edges between two nodes.
Line Graph A line graph G of a graph H is a graph with the
vertex set of E (H) and two vertices in G are
adjacent iff the corresponding edges in H are
adjacent.
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definitions and Observations
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Multigraph A multigraph H is a graph in which there can be
multiple edges between two nodes.
Line Graph A line graph G of a graph H is a graph with the
vertex set of E (H) and two vertices in G are
adjacent iff the corresponding edges in H are
adjacent.
Hypergraph A hypergraph is a graph in which an edge has as
endpoints two or more nodes. Every graph is the
line graph of a hypergraph.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definitions and Observations
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Multigraph A multigraph H is a graph in which there can be
multiple edges between two nodes.
Line Graph A line graph G of a graph H is a graph with the
vertex set of E (H) and two vertices in G are
adjacent iff the corresponding edges in H are
adjacent.
Hypergraph A hypergraph is a graph in which an edge has as
endpoints two or more nodes. Every graph is the
line graph of a hypergraph.
Multiplicity The multiplicity µ(a, b) of a pair of vertices a and
b are the number of edges between them.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Not every graph is the line graph of a multi graph
Reed’s
Conjecture
Counter example
Naveen
Sundar G.
Outline
Preliminaries
Hyper Graphs
Not every graph Figure:
is the line
graph of a multi graph
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
is the line graph
of a hyper graph
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Extending Vertex Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
A vertex coloring is equivalent to a set of stable sets or
independents sets.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Extending Vertex Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
A vertex coloring is equivalent to a set of stable sets or
independents sets.
2
Each stable set corresponds to one color
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Extending Vertex Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
A vertex coloring is equivalent to a set of stable sets or
independents sets.
2
Each stable set corresponds to one color
3
This can be generalized. Each vertex can be colored by
multiple colors with no two adjacent vertices sharing any
color.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Extending Vertex Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
A vertex coloring is equivalent to a set of stable sets or
independents sets.
2
Each stable set corresponds to one color
3
This can be generalized. Each vertex can be colored by
multiple colors with no two adjacent vertices sharing any
color.
4
Each color contributes fractionally. This gives rise to
fractional vertex coloring
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Extending Vertex Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
A vertex coloring is equivalent to a set of stable sets or
independents sets.
2
Each stable set corresponds to one color
3
This can be generalized. Each vertex can be colored by
multiple colors with no two adjacent vertices sharing any
color.
4
Each color contributes fractionally. This gives rise to
fractional vertex coloring
5
We can also set the colors available across different nodes
to be different. Each node has at its disposal the same
number of colors. This gives rise to list coloring
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definition of a Fractional Vertex Coloring
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
A fractional vertex c-coloring of a graph can be described as a
set S1 , . . . , Sl of stable sets with associated non-negative real
weights w1 , . . . , wn such that for vertex ν,
X
wi = 1
Si :ν∈Si
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
and
l
X
wi = c
i=1
The fractional chromatic number of G denoted by χf (G ) is the
smallest c for which G has a fractional vertex c coloring.
Naveen Sundar G.
Reed’s Conjecture
Reed’s Conjecture for Fractional Coloring
Reed’s
Conjecture
Naveen
Sundar G.
Outline
The conjecture holds for fractional vertex coloring
Preliminaries
Basic Results
The Conjecture
General
Properties
Theorem
For any graph G ,
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
χf (G ) ≤ d
∆(G ) + 1 + ω(G )
e
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
χ and χf
Reed’s
Conjecture
Naveen
Sundar G.
Every integer coloring is also a fractional coloring with wi = 1.
So we can never have have χ < χf . But is χf < χ possible?
Yes. The cycle of 5 vertices has χ = 3
Outline
Preliminaries
Figure: Coloring of a 5 Cycle
Basic Results
The Conjecture
General
Properties
a
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
1
1,4
2
3
1
2
Similar
Conjectures
b
e
3,5
2,5
4,2
3,1
d
Naveen Sundar G.
Reed’s Conjecture
c
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
The cycle of 5 vertices has 2.5 fractional coloring
A 2.5 Fractional Coloring
stable set
vertices
weights
S1
a,c
S2
b,d
S3
c,e
S4
d,a
S5
e,b
1
2
1
2
1
2
1
2
1
2
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Figure: Coloring of a 5 Cycle
a
1
1,4
2
3
1
2
b
e
3,5
2,5
4,2
3,1
d
Naveen Sundar G.
c
Reed’s Conjecture
Definition of List Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
Outline
We have to color each vertex from a predetermined list of
r colors.
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definition of List Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
We have to color each vertex from a predetermined list of
r colors.
2
The list may differ for each vertex. In normal coloring the
list is same across vertices.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definition of List Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
We have to color each vertex from a predetermined list of
r colors.
2
The list may differ for each vertex. In normal coloring the
list is same across vertices.
3
The smallest list size r such that no matter how we
choose the colors from each list the graph is properly
colorable is the list chromatic index χl .
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Definition of List Coloring
Reed’s
Conjecture
Naveen
Sundar G.
1
We have to color each vertex from a predetermined list of
r colors.
2
The list may differ for each vertex. In normal coloring the
list is same across vertices.
3
The smallest list size r such that no matter how we
choose the colors from each list the graph is properly
colorable is the list chromatic index χl .
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
4
We have χl definitely not less than χ. Can χl be greater
than χ?
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
List Coloring
Reed’s
Conjecture
Naveen
Sundar G.
List Coloring
from Erdos, ChoosabilityFigure:
in Graphs
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
[2,3]
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
[1,3]
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
List Coloring
[2,3]
[1,2]
Reed’s
Conjecture
Figure: List Coloring
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
1
2
[1,3]
[1,2]
[2,3]
[1,2]
3
[2,3]
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
?
Naveen Sundar G.
Reed’s Conjecture
[1,3]
1
[1,3]
List Coloring
[1,2]
[1,3]
Reed’s
Conjecture
Figure: List Coloring
Naveen
Sundar G.
3
Outline
Preliminaries
[2,3]
Basic Results
The Conjecture
General
Properties
3
?
[1,3]
[1,2]
[2,3]
[1,2]
2
[2,3]
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
[1,3]
Similar
1
Conjectures
2
Naveen Sundar G.
1
Reed’s Conjecture
[1,3]
3
Chromatic Index Xe (H)
Reed’s
Conjecture
Naveen
Sundar G.
Outline
1
The chromatic index χe (H) of a hypergraph H is the
chromatic number χ(G ) of its line graph(G = L(H).)
2
The following theorem by Caprara and Rizzi puts an upper
bound on the chromatic index of a multigraph
χe (H) ≤ max(b1.1∆(H) + 0.7c, dΓ(H)e).
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
3
Goldberg-Seymour Conjecture For a multigraph H for
which χe (H) > ∆(H) + 1 , χe (H) = dΓ(H)e
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Fractional Chromatic Index Xe (H)
Reed’s
Conjecture
Naveen
Sundar G.
1
The fractional chromatic index χfe (H) of a hypergraph H
is the fractional chromatic number χ(G ) of its line graph
χf (G ) (G = L(H).)
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Fractional Chromatic Index Xe (H)
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
The fractional chromatic index χfe (H) of a hypergraph H
is the fractional chromatic number χ(G ) of its line graph
χf (G ) (G = L(H).)
A matching in H corresponds to a stable set in G .
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Fractional Chromatic Index Xe (H)
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
Preliminaries
3
Basic Results
The Conjecture
General
Properties
Line Graphs
The fractional chromatic index χfe (H) of a hypergraph H
is the fractional chromatic number χ(G ) of its line graph
χf (G ) (G = L(H).)
A matching in H corresponds to a stable set in G .
Given a nonnegative weighting w on the edges
P of H such
that for every matching
M
in
H
we
have
e∈M w (e) ≤ 1
P
then χfe (H) ≥ e∈E (H) w (e)
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Fractional Chromatic Index Xe (H)
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
Preliminaries
3
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
The fractional chromatic index χfe (H) of a hypergraph H
is the fractional chromatic number χ(G ) of its line graph
χf (G ) (G = L(H).)
A matching in H corresponds to a stable set in G .
Given a nonnegative weighting w on the edges
P of H such
that for every matching
M
in
H
we
have
e∈M w (e) ≤ 1
P
then χfe (H) ≥ e∈E (H) w (e)
Proof.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Fractional Chromatic Index Xe (H)
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
Preliminaries
3
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
The fractional chromatic index χfe (H) of a hypergraph H
is the fractional chromatic number χ(G ) of its line graph
χf (G ) (G = L(H).)
A matching in H corresponds to a stable set in G .
Given a nonnegative weighting w on the edges
P of H such
that for every matching
M
in
H
we
have
e∈M w (e) ≤ 1
P
then χfe (H) ≥ e∈E (H) w (e)
Proof.
1
χfe (H) = χf (G ) =
wi
ν∈Si |Si |
P P
ν
such that
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
P
i:ν∈Si
wi = 1
Fractional Chromatic Index Xe (H)
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
Preliminaries
3
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
The fractional chromatic index χfe (H) of a hypergraph H
is the fractional chromatic number χ(G ) of its line graph
χf (G ) (G = L(H).)
A matching in H corresponds to a stable set in G .
Given a nonnegative weighting w on the edges
P of H such
that for every matching
M
in
H
we
have
e∈M w (e) ≤ 1
P
then χfe (H) ≥ e∈E (H) w (e)
Proof.
1
2
P
P P
χfe (H) = χf (G ) = ν ν∈Si |Swii | such that i:ν∈Si wi = 1
P
P
P P
wi
i:ν∈Si wi ≤ 1
e∈E (H) w (e) =
ν
ν∈Si |Si | such that
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Fractional Chromatic Index Xe (H)
Reed’s
Conjecture
1
Naveen
Sundar G.
Outline
2
Preliminaries
3
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
The fractional chromatic index χfe (H) of a hypergraph H
is the fractional chromatic number χ(G ) of its line graph
χf (G ) (G = L(H).)
A matching in H corresponds to a stable set in G .
Given a nonnegative weighting w on the edges
P of H such
that for every matching
M
in
H
we
have
e∈M w (e) ≤ 1
P
then χfe (H) ≥ e∈E (H) w (e)
Proof.
1
2
3
P
P P
χfe (H) = χf (G ) = ν ν∈Si |Swii | such that i:ν∈Si wi = 1
P
P
P P
wi
i:ν∈Si wi ≤ 1
e∈E (H) w (e) =
ν
ν∈Si |Si | such that
Both the sums over the same set of indices.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Bounds on the fractional chromatic index
Reed’s
Conjecture
Naveen
Sundar G.
1
We consider two such weightings.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Bounds on the fractional chromatic index
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
1
We consider two such weightings.
1
A weight of 1 to each edge incident to just one vertex of
maximum degree. Every other edge is assigned a weight 0.
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Bounds on the fractional chromatic index
Reed’s
Conjecture
Naveen
Sundar G.
Outline
1
We consider two such weightings.
1
Preliminaries
Basic Results
The Conjecture
General
Properties
2
A weight of 1 to each edge incident to just one vertex of
maximum degree. Every other edge is assigned a weight 0.
We take an induced subgraph W of H and assign to each
edge of W a weight of 1/b|V (W )|/2c and other edges 0
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Bounds on the fractional chromatic index
Reed’s
Conjecture
Naveen
Sundar G.
1
Outline
We consider two such weightings.
1
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
2
2
A weight of 1 to each edge incident to just one vertex of
maximum degree. Every other edge is assigned a weight 0.
We take an induced subgraph W of H and assign to each
edge of W a weight of 1/b|V (W )|/2c and other edges 0
We can derive χfe (H) ≤ max(∆(H), Γ(H))
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Bounds on the fractional chromatic index
Reed’s
Conjecture
Naveen
Sundar G.
1
Outline
We consider two such weightings.
1
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
2
2
3
A weight of 1 to each edge incident to just one vertex of
maximum degree. Every other edge is assigned a weight 0.
We take an induced subgraph W of H and assign to each
edge of W a weight of 1/b|V (W )|/2c and other edges 0
We can derive χfe (H) ≤ max(∆(H), Γ(H))
(W )|
where Γ(H) = max |V2|E
(W )−1| : W ⊂ H, |V (W )|is odd
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Bounds on the fractional chromatic index
Reed’s
Conjecture
Naveen
Sundar G.
1
Outline
We consider two such weightings.
1
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
2
2
3
4
A weight of 1 to each edge incident to just one vertex of
maximum degree. Every other edge is assigned a weight 0.
We take an induced subgraph W of H and assign to each
edge of W a weight of 1/b|V (W )|/2c and other edges 0
We can derive χfe (H) ≤ max(∆(H), Γ(H))
(W )|
where Γ(H) = max |V2|E
(W )−1| : W ⊂ H, |V (W )|is odd
Edmond’s theorem for matching polytypes
χfe (H) = max(∆(H), Γ(H))
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Reed’s Conjecture for Line Graphs
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Theorem
For any line graph G , Reed’s conjecture holds.
We set G = L(H) and consider two cases
1
2
∆(G ) is large. That is ∆(G ) ≥ 23 ∆(H) − 1
∆(G ) is small. That is ∆(G ) < 32 ∆(H) − 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is large
Reed’s
Conjecture
Naveen
Sundar G.
Proof.
1
Given: χfe = max(∆(H), Γ(H))
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is large
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Proof.
1
2
Given: χfe = max(∆(H), Γ(H))
Given: For any multigraph
χe (H) ≤ max(b1.1∆(H) + 0.7c, dΓ(H)e). (Caprara)
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is large
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Proof.
1
2
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
3
Given: χfe = max(∆(H), Γ(H))
Given: For any multigraph
χe (H) ≤ max(b1.1∆(H) + 0.7c, dΓ(H)e). (Caprara)
Combining 1 and 2 we get
χ(G ) ≤ max(b1.1∆(H) + 0.7c, dχf (G )e)
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is large
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Proof.
1
2
Preliminaries
Basic Results
The Conjecture
General
Properties
3
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
4
Given: χfe = max(∆(H), Γ(H))
Given: For any multigraph
χe (H) ≤ max(b1.1∆(H) + 0.7c, dΓ(H)e). (Caprara)
Combining 1 and 2 we get
χ(G ) ≤ max(b1.1∆(H) + 0.7c, dχf (G )e)
Using Reed’s theorem for fractional coloring
)
χ(G ) ≤ max(b1.1∆(H) + 0.7c, d ∆(G )+1+ω(G
e)
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is large
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Proof.
1
2
Preliminaries
Basic Results
The Conjecture
General
Properties
3
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
4
5
Given: χfe = max(∆(H), Γ(H))
Given: For any multigraph
χe (H) ≤ max(b1.1∆(H) + 0.7c, dΓ(H)e). (Caprara)
Combining 1 and 2 we get
χ(G ) ≤ max(b1.1∆(H) + 0.7c, dχf (G )e)
Using Reed’s theorem for fractional coloring
)
χ(G ) ≤ max(b1.1∆(H) + 0.7c, d ∆(G )+1+ω(G
e)
2
Assumption: ∆(G ) ≥ 23 ∆(H) − 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is large
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Proof.
1
2
Preliminaries
Basic Results
The Conjecture
General
Properties
3
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
4
5
6
Given: χfe = max(∆(H), Γ(H))
Given: For any multigraph
χe (H) ≤ max(b1.1∆(H) + 0.7c, dΓ(H)e). (Caprara)
Combining 1 and 2 we get
χ(G ) ≤ max(b1.1∆(H) + 0.7c, dχf (G )e)
Using Reed’s theorem for fractional coloring
)
χ(G ) ≤ max(b1.1∆(H) + 0.7c, d ∆(G )+1+ω(G
e)
2
Assumption: ∆(G ) ≥ 23 ∆(H) − 1
)
Therefore d ∆(G )+1+ω(G
e ≥ d 54 e ≥ b1.1∆(H) + 0.7c
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is large
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Proof.
1
2
Preliminaries
Basic Results
The Conjecture
General
Properties
3
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
4
5
6
7
Given: χfe = max(∆(H), Γ(H))
Given: For any multigraph
χe (H) ≤ max(b1.1∆(H) + 0.7c, dΓ(H)e). (Caprara)
Combining 1 and 2 we get
χ(G ) ≤ max(b1.1∆(H) + 0.7c, dχf (G )e)
Using Reed’s theorem for fractional coloring
)
χ(G ) ≤ max(b1.1∆(H) + 0.7c, d ∆(G )+1+ω(G
e)
2
Assumption: ∆(G ) ≥ 23 ∆(H) − 1
)
Therefore d ∆(G )+1+ω(G
e ≥ d 54 e ≥ b1.1∆(H) + 0.7c
2
Therefore χ(G ) ≤
∆(G )+1ω(G )
2
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is small
Reed’s
Conjecture
Given
Naveen
Sundar G.
∆(G ) < 32 ∆(H) − 1
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
We prove using induction. The base case consisting of all the
hypergraph of two vertices satisfies Reed’s conjecture.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is small
Reed’s
Conjecture
Given
Naveen
Sundar G.
∆(G ) < 32 ∆(H) − 1
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
We prove using induction. The base case consisting of all the
hypergraph of two vertices satisfies Reed’s conjecture.
1 G 0 is the subgraph of G with fewer vertices which needs to
satisfy ∆(G 0 ) ≤ ∆(G ) − 1 (since S is maximal) and
ω(G 0 ) = ω(G ) − 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is small
Reed’s
Conjecture
Given
Naveen
Sundar G.
∆(G ) < 32 ∆(H) − 1
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
We prove using induction. The base case consisting of all the
hypergraph of two vertices satisfies Reed’s conjecture.
1 G 0 is the subgraph of G with fewer vertices which needs to
satisfy ∆(G 0 ) ≤ ∆(G ) − 1 (since S is maximal) and
ω(G 0 ) = ω(G ) − 1
2 G 0 is also a line graph and we have using the induction
hypothesis and (2) and (3)
χ(G 0 ) ≤ d
Naveen Sundar G.
∆(G 0 ) + 1 + ω(G 0 )
e
2
Reed’s Conjecture
Proof when ∆(G ) is small
Reed’s
Conjecture
Given
Naveen
Sundar G.
∆(G ) < 32 ∆(H) − 1
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
We prove using induction. The base case consisting of all the
hypergraph of two vertices satisfies Reed’s conjecture.
1 G 0 is the subgraph of G with fewer vertices which needs to
satisfy ∆(G 0 ) ≤ ∆(G ) − 1 (since S is maximal) and
ω(G 0 ) = ω(G ) − 1
2 G 0 is also a line graph and we have using the induction
hypothesis and (2) and (3)
χ(G 0 ) ≤ d
Similar
Conjectures
3
∆(G 0 ) + 1 + ω(G 0 )
e
2
)
combining the above χ(G 0 ) ≤ d ∆(G )+1+ω(G
e−1
2
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is small
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
4
V (G ) \ V (G 0 ) should be a stable set
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is small
Reed’s
Conjecture
Naveen
Sundar G.
Outline
4
Preliminaries
5
Basic Results
The Conjecture
General
Properties
Line Graphs
V (G ) \ V (G 0 ) should be a stable set
We can construct a proper χ(G 0 ) + 1-coloring of V (G ) by
taking the χ(G 0 ) coloring of G 0 and we take S to be the
final color class.
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is small
Reed’s
Conjecture
Naveen
Sundar G.
Outline
4
Preliminaries
5
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
6
V (G ) \ V (G 0 ) should be a stable set
We can construct a proper χ(G 0 ) + 1-coloring of V (G ) by
taking the χ(G 0 ) coloring of G 0 and we take S to be the
final color class.
Therefore χ(G ) ≤
∆(G )+1+ω(G )
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Proof when ∆(G ) is small
Reed’s
Conjecture
Naveen
Sundar G.
Outline
4
Preliminaries
5
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
6
7
V (G ) \ V (G 0 ) should be a stable set
We can construct a proper χ(G 0 ) + 1-coloring of V (G ) by
taking the χ(G 0 ) coloring of G 0 and we take S to be the
final color class.
Therefore χ(G ) ≤
∆(G )+1+ω(G )
2
Find a maximal stable set S ⊂ V (G ) that has a vertex
from every maximum clique in G .
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Basic Idea
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
We need to show when ∆(G ) < 23 ∆(H) − 1 we have a maximal
stable set S which contains a vertex from every maximum
clique.
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Basic Idea
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
We need to show when ∆(G ) < 23 ∆(H) − 1 we have a maximal
stable set S which contains a vertex from every maximum
clique.
1
∆(G ) = maxuv ∈E (H) {deg (u) + deg (v ) − µ(u, v ) − 1}
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Basic Idea
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
We need to show when ∆(G ) < 23 ∆(H) − 1 we have a maximal
stable set S which contains a vertex from every maximum
clique.
1
2
∆(G ) = maxuv ∈E (H) {deg (u) + deg (v ) − µ(u, v ) − 1}
Every maximum clique in G comes either from a vertex of
maximum degree in H or a triangle with a large number of
edges in H.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Basic Idea
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
We need to show when ∆(G ) < 23 ∆(H) − 1 we have a maximal
stable set S which contains a vertex from every maximum
clique.
1
∆(G ) = maxuv ∈E (H) {deg (u) + deg (v ) − µ(u, v ) − 1}
2
Every maximum clique in G comes either from a vertex of
maximum degree in H or a triangle with a large number of
edges in H.
3
If tri(H) is the maximum number of edges in a triangle.
ω(G ) = max{∆(H), tri(H)}
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Basic Idea Continued
Reed’s
Conjecture
Naveen
Sundar G.
1
Outline
We say that a matching hits a vertex v if it is an endpoint
of an edge in the matching.
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Basic Idea Continued
Reed’s
Conjecture
Naveen
Sundar G.
1
We say that a matching hits a vertex v if it is an endpoint
of an edge in the matching.
2
We need to find a maximal matching M in H which will
correspond to the stable set S in G . M should hit every
vertex of maximum degree in H and contain an edge of
every triangle with max{∆(H), tri(H)} edges.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Basic Idea Continued
Reed’s
Conjecture
Naveen
Sundar G.
1
We say that a matching hits a vertex v if it is an endpoint
of an edge in the matching.
2
We need to find a maximal matching M in H which will
correspond to the stable set S in G . M should hit every
vertex of maximum degree in H and contain an edge of
every triangle with max{∆(H), tri(H)} edges.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
S∆ is the set of vertices in H of degree ∆(H)
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Basic Idea Continued
Reed’s
Conjecture
Naveen
Sundar G.
1
We say that a matching hits a vertex v if it is an endpoint
of an edge in the matching.
2
We need to find a maximal matching M in H which will
correspond to the stable set S in G . M should hit every
vertex of maximum degree in H and contain an edge of
every triangle with max{∆(H), tri(H)} edges.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
S∆ is the set of vertices in H of degree ∆(H)
T is the set of all triangles in H with max{∆(H), tri(H)}
edges.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 1
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Theorem (Lemma 1)
If two triangles of T intersect in exactly the vertices a and b
then ab has multiplicity greater than ∆(H)/2
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 1
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Theorem (Lemma 1)
If two triangles of T intersect in exactly the vertices a and b
then ab has multiplicity greater than ∆(H)/2
Preliminaries
Basic Results
The Conjecture
General
Properties
Proof.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 1
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Theorem (Lemma 1)
If two triangles of T intersect in exactly the vertices a and b
then ab has multiplicity greater than ∆(H)/2
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
3
3
∆H > ∆(H) − 1 > ∆(G ) ≥ 2∆(H) − µ(a, b)
2
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 1
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Theorem (Lemma 1)
If two triangles of T intersect in exactly the vertices a and b
then ab has multiplicity greater than ∆(H)/2
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
3
3
∆H > ∆(H) − 1 > ∆(G ) ≥ 2∆(H) − µ(a, b)
2
2
3
∆H > 2∆(H) − µ(a, b)
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 1
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Theorem (Lemma 1)
If two triangles of T intersect in exactly the vertices a and b
then ab has multiplicity greater than ∆(H)/2
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Proof.
3
3
∆H > ∆(H) − 1 > ∆(G ) ≥ 2∆(H) − µ(a, b)
2
2
3
∆H > 2∆(H) − µ(a, b)
2
µ(a, b) ≥ ∆(H)/2
Naveen Sundar G.
Reed’s Conjecture
Lemma 2
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 2)
If abc is a triangle of T intersecting another triangle ade of T
in exactly the vertex a then µ(b, c) is greater than ∆(H)/2
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 2
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 2)
If abc is a triangle of T intersecting another triangle ade of T
in exactly the vertex a then µ(b, c) is greater than ∆(H)/2
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Proof.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 2
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 2)
If abc is a triangle of T intersecting another triangle ade of T
in exactly the vertex a then µ(b, c) is greater than ∆(H)/2
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
The degree of a vertex of G corresponding to an edge between
a and d is at least 2∆(H) − µ(b, c) − 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 2
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 2)
If abc is a triangle of T intersecting another triangle ade of T
in exactly the vertex a then µ(b, c) is greater than ∆(H)/2
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
The degree of a vertex of G corresponding to an edge between
a and d is at least 2∆(H) − µ(b, c) − 1
3
∆(H) − 1 > ∆(G ) ≥ 2∆(H) − µ(b, c) − 1
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 2
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 2)
If abc is a triangle of T intersecting another triangle ade of T
in exactly the vertex a then µ(b, c) is greater than ∆(H)/2
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
The degree of a vertex of G corresponding to an edge between
a and d is at least 2∆(H) − µ(b, c) − 1
3
∆(H) − 1 > ∆(G ) ≥ 2∆(H) − µ(b, c) − 1
2
3
∆(H) − 1 > 2∆(H) − µ(b, c) − 1
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 2
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 2)
If abc is a triangle of T intersecting another triangle ade of T
in exactly the vertex a then µ(b, c) is greater than ∆(H)/2
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Proof.
The degree of a vertex of G corresponding to an edge between
a and d is at least 2∆(H) − µ(b, c) − 1
3
∆(H) − 1 > ∆(G ) ≥ 2∆(H) − µ(b, c) − 1
2
3
∆(H) − 1 > 2∆(H) − µ(b, c) − 1
2
1
∴ µ(b, c) ≥ ∆(H)
2
Naveen Sundar G.
Reed’s Conjecture
Lemma 3
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Theorem (Lemma 3)
If there is an edge of H joining two vertices a and b of S∆ then
µ(a, b) > ∆(H)/2
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 3
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Theorem (Lemma 3)
If there is an edge of H joining two vertices a and b of S∆ then
µ(a, b) > ∆(H)/2
Proof.
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 3
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Theorem (Lemma 3)
If there is an edge of H joining two vertices a and b of S∆ then
µ(a, b) > ∆(H)/2
Proof.
3
∆(H) − 1 > ∆(G ) ≥ 2∆(H) − µ(a, b) − 1
2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 3
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Theorem (Lemma 3)
If there is an edge of H joining two vertices a and b of S∆ then
µ(a, b) > ∆(H)/2
Proof.
3
∆(H) − 1 > ∆(G ) ≥ 2∆(H) − µ(a, b) − 1
2
∴ u(a, b) > ∆(H)/2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
More Terminology
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
1
T 0 is the set of triangles in T that contain no pair of
vertices of multiplicity > ∆(H)/2
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
More Terminology
Reed’s
Conjecture
Naveen
Sundar G.
Outline
1
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
2
T 0 is the set of triangles in T that contain no pair of
vertices of multiplicity > ∆(H)/2
0 are those elements of S which are not part of any
S∆
∆
pair of vertices of multiplicity > ∆(H)/2
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
More Terminology
Reed’s
Conjecture
Naveen
Sundar G.
Outline
1
Preliminaries
Basic Results
The Conjecture
General
Properties
2
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
3
T 0 is the set of triangles in T that contain no pair of
vertices of multiplicity > ∆(H)/2
0 are those elements of S which are not part of any
S∆
∆
pair of vertices of multiplicity > ∆(H)/2
For a set of vertices S the union of the vertices’
neighbourhoods is N(S)
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 4
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 4)
0 we have |N(S)| ≥ |S|
For any S ⊂ S∆
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 4
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 4)
0 we have |N(S)| ≥ |S|
For any S ⊂ S∆
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Proof.
0 that is a stable set. This
It follows from Lemma 3 that S∆
implies that S and N(S) are disjoint.
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 4
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 4)
0 we have |N(S)| ≥ |S|
For any S ⊂ S∆
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
0 that is a stable set. This
It follows from Lemma 3 that S∆
implies that S and N(S) are disjoint. There are |S|∆(H) edges
between S and N(S).
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 4
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 4)
0 we have |N(S)| ≥ |S|
For any S ⊂ S∆
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Proof.
0 that is a stable set. This
It follows from Lemma 3 that S∆
implies that S and N(S) are disjoint. There are |S|∆(H) edges
between S and N(S). Upper bound on the number of edges
from nodes in N(S) is N(S)∆(H)
N(S)∆(H) ≥ |S|∆(H)
∴ |N(S)| ≥ |S|
Naveen Sundar G.
Reed’s Conjecture
Lemma 5
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 5)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 , then the degree of a is less than ∆(H)
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 5
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 5)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 , then the degree of a is less than ∆(H)
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Proof.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 5
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 5)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 , then the degree of a is less than ∆(H)
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
Any vertex in G corresponding to an edge between a and b has
degree at least deg (a) − 1 + ∆(H) − µ(c, d)
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 5
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 5)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 , then the degree of a is less than ∆(H)
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
Any vertex in G corresponding to an edge between a and b has
degree at least deg (a) − 1 + ∆(H) − µ(c, d)
µ(c, d) ≤ ∆(H) + 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 5
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 5)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 , then the degree of a is less than ∆(H)
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Proof.
Any vertex in G corresponding to an edge between a and b has
degree at least deg (a) − 1 + ∆(H) − µ(c, d)
µ(c, d) ≤ ∆(H) + 1
3/2∆(H) − 1 ≥ deg (a) − 1 + ∆(H)/2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 5
Reed’s
Conjecture
Naveen
Sundar G.
Theorem (Lemma 5)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 , then the degree of a is less than ∆(H)
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Proof.
Any vertex in G corresponding to an edge between a and b has
degree at least deg (a) − 1 + ∆(H) − µ(c, d)
µ(c, d) ≤ ∆(H) + 1
3/2∆(H) − 1 ≥ deg (a) − 1 + ∆(H)/2
∴ ∆(H)/2 ≥ deg (a)
Naveen Sundar G.
Reed’s Conjecture
Lemma 6
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Theorem (Lemma 6)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 then µ(a, b) ≤ ∆(H)/2
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 6
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Theorem (Lemma 6)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 then µ(a, b) ≤ ∆(H)/2
Proof.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 6
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Theorem (Lemma 6)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 then µ(a, b) ≤ ∆(H)/2
Proof.
The degree of any vertex in G corresponding to an edge
between b and c has degree at least µ(a, b) + ∆(H) − 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 6
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Theorem (Lemma 6)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 then µ(a, b) ≤ ∆(H)/2
Proof.
The degree of any vertex in G corresponding to an edge
between b and c has degree at least µ(a, b) + ∆(H) − 1
3/2∆(H) − 1 ≥ µ(a, b) + ∆(H) − 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 6
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Theorem (Lemma 6)
If an edge ab in H has exactly one endpoint in a triangle bcd
of T 0 then µ(a, b) ≤ ∆(H)/2
Proof.
The degree of any vertex in G corresponding to an edge
between b and c has degree at least µ(a, b) + ∆(H) − 1
3/2∆(H) − 1 ≥ µ(a, b) + ∆(H) − 1
∴ µ(a, b) ≤ ∆(H)/2
Naveen Sundar G.
Reed’s Conjecture
Lemma 7
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Theorem (Lemma 7)
For any vertex v with neighbors v and w ,
deg (u) + µ(vw ) − 1 ≤ 3/2∆(H) − 1
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 7
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Theorem (Lemma 7)
For any vertex v with neighbors v and w ,
deg (u) + µ(vw ) − 1 ≤ 3/2∆(H) − 1
Proof.
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 7
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Theorem (Lemma 7)
For any vertex v with neighbors v and w ,
deg (u) + µ(vw ) − 1 ≤ 3/2∆(H) − 1
Proof.
An edge between u and v is incident to at least
deg (u) + µ(vw ) − 1 other edges
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 7
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Theorem (Lemma 7)
For any vertex v with neighbors v and w ,
deg (u) + µ(vw ) − 1 ≤ 3/2∆(H) − 1
Proof.
An edge between u and v is incident to at least
deg (u) + µ(vw ) − 1 other edges
(u) + µ(vw ) − 1 ≤ 3/2∆(H) − 1
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Lemma 7
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Theorem (Lemma 7)
For any vertex v with neighbors v and w ,
deg (u) + µ(vw ) − 1 ≤ 3/2∆(H) − 1
Proof.
An edge between u and v is incident to at least
deg (u) + µ(vw ) − 1 other edges
(u) + µ(vw ) − 1 ≤ 3/2∆(H) − 1
∴ deg (u) + µ(vw ) ≤ 3/2∆(H) − 1
Naveen Sundar G.
Reed’s Conjecture
Hall’s Theorem
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Theorem
Let G be a bipartite graph with vertex set V = (A,B). There is
a matching that hits every vertex in A precisely if for every
S ⊂ A we have |N(S)| ≥ |S|
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Completion of the proof
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
We need to show that our desired matching exists,
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Completion of the proof
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
We need to show that our desired matching exists,
We construct three matchings and combine them. Have to
show the combination is still a matching and the combination
is possible.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Construction of a matching
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Step one: Construct matching M1
One edge between each vertex pair with multiplicity greater
0 and contains one edge of each
than ∆(H)/2. This hits S∆ \ S∆
triangle in T \ T 0
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Construction of a matching
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Step two: Construct matching M2
0 we have |N(S)| ≥ |S|) and
Using Lemma 4 (For any S ⊂ S∆
0
Hall’s theorem we can construct a matching that hits S∆
Lemma 7 shows that this matching cannot hit M2 so the union
on M1 and M2 hits S∆ and an edge of each triangle in T \ T 0
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Construction of a matching
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Step two: Construct matching M2
0 we have |N(S)| ≥ |S|) and
Using Lemma 4 (For any S ⊂ S∆
0
Hall’s theorem we can construct a matching that hits S∆
Lemma 7 shows that this matching cannot hit M2 so the union
on M1 and M2 hits S∆ and an edge of each triangle in T \ T 0
Every edge in the matching M 0 = M1 ∪ M2 hits a
maximum-vertex in H or has endpoints with multiplicity greater
than ∆(H)/2
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Figure: The Final Matching
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
deg(u) = ∆(H)
u
!
S∆
M2
deg(u) + deg(vw) ≤ 3/2∆(H)
v
M1
µ(a, b) > ∆(H)/2
w
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Construction of a matching
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Using lemma 4 and We have to include edges from T 0 . We can
blindly add an arbitrary edge from each triangle in T 0 but at
least two vertices should remain uncontaminated by M 0
Preliminaries
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Step three: Construct matching M
Lemmas 3 ,5 and 6 show that M 0 hits at most one vertex in
T 0 . We extend M 0 to contain an edge of every triangle in T 0
and obtain M
This matching M satisfies our requirements. This matching
corresponds to our maximal stable set in G that has a vertex
from every maximum clique.
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Reed’s
Conjecture
Figure: The Final Matching
Naveen
Sundar G.
µ ≤ ∆(2)/2
Preliminaries
Basic Results
The Conjecture
General
Properties
Ruled out by Lemmas 5
Ruled out by Lemma 6
Outline
∆(H)
µ > ∆(2)/2
M!
µ > ∆(2)/2
M!
M!
∆(H)
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
µ ≤ ∆(2)/2
M!
∆(H)
Not possible
µ ≤ ∆(2)/2
µ ≤ ∆(2)/2
Only one node can be
of maximum degree
Only possible hitting
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Similar Conjectures
Reed’s
Conjecture
Naveen
Sundar G.
Outline
1
Preliminaries
Basic Results
The Conjecture
General
Properties
χ(G ) ≤
ω(G )+∆(G )
2
+ o(ω(G ))
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Similar Conjectures
Reed’s
Conjecture
Naveen
Sundar G.
Outline
1
χ(G ) ≤
2
χ(G ) ≤
Preliminaries
Basic Results
The Conjecture
General
Properties
ω(G )+∆(G )
2
ω(G )+∆(G )
2
+ o(ω(G ))
+ o(∆(G ))
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Similar Conjectures
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
ω(G )+∆(G )
2
ω(G )+∆(G )
2
1
χ(G ) ≤
2
χ(G ) ≤
3
with ∆(G ) ≥ 3 χ(G ) ≤
Preliminaries
+ o(ω(G ))
+ o(∆(G ))
2(∆(G )+1)+ω(G )
3
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture
Similar Conjectures
Reed’s
Conjecture
Naveen
Sundar G.
Outline
Basic Results
The Conjecture
General
Properties
Line Graphs
Definitions and
Observations
Fractional
vertex c-coloring
List Coloring
Chromatic Index
and Fractional
Chromatic Index
The Theorem
and Proof
Case I
Case II
ω(G )+∆(G )
2
ω(G )+∆(G )
2
1
χ(G ) ≤
2
χ(G ) ≤
3
with ∆(G ) ≥ 3 χ(G ) ≤
Preliminaries
4
+ o(ω(G ))
+ o(∆(G ))
2(∆(G )+1)+ω(G )
3
There is some constant α such that for any graph
χ(G ) ≤ αω(G ) + 12 (∆(G ) + 1)
Similar
Conjectures
Naveen Sundar G.
Reed’s Conjecture