Definitions
Bipartite Graphs
Section 1.3: Edge Counting
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Definitions
Bipartite Graphs
Theorem 1
Thm 1 In any graph, the sum of the degrees of all vertices is
equal to twice the number of edges.
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Definitions
Bipartite Graphs
Theorem 1
Thm 1 In any graph, the sum of the degrees of all vertices is
equal to twice the number of edges.
Proof: Summing the degrees of all vertices counts all
instances of some edge being incident at some
vertex. But each edge is incident with two vertices
and so the total number of such edge-vertex
incidences is simply twice the number of edges.
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Definitions
Bipartite Graphs
Theorem 1
Thm 1 In any graph, the sum of the degrees of all vertices is
equal to twice the number of edges.
Proof: Summing the degrees of all vertices counts all
instances of some edge being incident at some
vertex. But each edge is incident with two vertices
and so the total number of such edge-vertex
incidences is simply twice the number of edges.
Corr In any graph, the number of vertices of odd degree is
even.
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Definitions
Bipartite Graphs
Complete Graphs
A graph with n vertices in which each vertex is adjacent to all the
other vertices is called a complete graph on n vertices, denoted Kn .
Figure:
Image retrieved from
http://www.google.com/imgres?imgurl=http://mathworld.wolfram.com/images/eps-gif/CompleteGraphs
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Definitions
Bipartite Graphs
Definition
A graph G is bipartite if its vertices can be partitioned into two sets
V1 and V2 and every edge joins a vertex in V1 with a vertex in V2 .
Figure:
Image retrieved from
http://leizhangmth350.wordpress.com/2012/09/29/graph-theory-crossing-numbers-of-bipartite-graphs/
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Definitions
Bipartite Graphs
Theorem 2
A graph G is bipartite if and only if every circuit in G has even
length.
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Definitions
Bipartite Graphs
Theorem 2
A graph G is bipartite if and only if every circuit in G has even
length.
Sketch of Proof: Suppose G is bipartite. This means that the
vertices of G can be partitioned into two sets V and U. WLOG
choose a cycle such as v1 − u1 − v2 − u2 − · · · − vn − un − v1 where
vi ∈ V and ui ∈ U. There are 2n vertices in this cycle.
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Definitions
Bipartite Graphs
Theorem 2
A graph G is bipartite if and only if every circuit in G has even
length.
Sketch of Proof: Suppose G is bipartite. This means that the
vertices of G can be partitioned into two sets V and U. WLOG
choose a cycle such as v1 − u1 − v2 − u2 − · · · − vn − un − v1 where
vi ∈ V and ui ∈ U. There are 2n vertices in this cycle.
Suppose every circuit in G has even length. WLOG choose a
vertex u. Every cycle which passes through u has even length.
Every vertex which is an even number of steps from u is in U and
every vertex which is an odd number of steps from u is in V . You
can extend this labeling of vertices to every cycle which intersects
this cycle because it also has even length. If the graph is not
connected you can do the same thing in the other components.
Then U and V partition the vertices and G is a bipartite graph.
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Definitions
Bipartite Graphs
Testing for a Bipartite Graph
Figure:
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Definitions
Bipartite Graphs
Testing for a Bipartite Graph
Figure:
Image retrieved from http://users.dickinson.edu/ braught/courses/cs332s03/projects/project2.html
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Definitions
Bipartite Graphs
Complete Bipartite Graph
The complete bipartite graph Km,n is defined by taking two disjoint
sets, V1 of size m and V2 of size n, and putting an edge between u
and v whenever u ∈ V1 and v ∈ V2 .
Figure:
Image retrieved from http://www.southernct.edu/ fields/comb dic/P.html
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