Bipartite graphs Consider the following graph G, edge set of G1 is E = {(v1, v5), (V1, v6), (v2, v3), (v2, V7), (v3, v5), (V4, V5), (v5, v7), (V6, V7)}. Graph G1 thus has n = |V|=7 vertices and m = |E| = 8 edges. (V, E). The vertex set of G1 is V = {v1, v2, V3, V4, V5, V6, V7}, and the %3D 5 3 Let A = {v1, v3, v4, v7} and B = {v2, v5, v6}. Observe that (1) A C V and B C V are subsets of V, (2) their union AUB = V contains all of the vertices in V, and (3) the intersection ANB= 0 is the empty set. Also observe that for every edge e e E, it holds that one of the endpoints of e is in A and the other endpoint is in B. We say that a graph G = (V, E) is bipartite if there exist subsets A C V and BC V such that AUB = V and ANB= 0, and for every edge in E, edge e has one endpoint in A and one endpoint in B. The graph G, given above is bipartite. 1. For each of the following four graphs, determine whether it is bipartite or not. Justify your answer.

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bipartitr graphs

Bipartite graphs
Consider the following graph G¡ = (V, E). The vertex set of G1 is V = {v1, v2, V3, V4, V5, V6, V7}, and the
edge set of G1 is E = {(v1, v5), (v1, v6), (v2, v3), (V2, v7), (v3, v5), (v4, v5), (V5, v7), (V6, V7)}. Graph G1 thus has
n = |V|= 7 vertices and m = |E| = 8 edges.
Let A = {v1, v3, V4, V7} and B = {v2, v5, v6}. Observe that (1) ACV and B C V are subsets of V, (2) their
union AU B =V contains all of the vertices in V, and (3) the intersection AN B = Ø is the empty set. Also
observe that for every edge e € E, it holds that one of the endpoints of e is in A and the other endpoint is
in B.
We say that a graph G = (V, E) is bipartite if there exist subsets A C V and B C V such that AUB = V
and ANB=0, and for every edge in E, edge e has one endpoint in A and one endpoint in B. The graph G1
given above is bipartite.
1. For each of the following four graphs, determine whether it is bipartite or not. Justify your answer.
2. Show that if a graph G = (V, E) is bipartite, then it has at most n? edges if n is even, and at most
|(n2 – 1) edges if n is odd. Here n = |V| is the number of vertices in G.
Transcribed Image Text:Bipartite graphs Consider the following graph G¡ = (V, E). The vertex set of G1 is V = {v1, v2, V3, V4, V5, V6, V7}, and the edge set of G1 is E = {(v1, v5), (v1, v6), (v2, v3), (V2, v7), (v3, v5), (v4, v5), (V5, v7), (V6, V7)}. Graph G1 thus has n = |V|= 7 vertices and m = |E| = 8 edges. Let A = {v1, v3, V4, V7} and B = {v2, v5, v6}. Observe that (1) ACV and B C V are subsets of V, (2) their union AU B =V contains all of the vertices in V, and (3) the intersection AN B = Ø is the empty set. Also observe that for every edge e € E, it holds that one of the endpoints of e is in A and the other endpoint is in B. We say that a graph G = (V, E) is bipartite if there exist subsets A C V and B C V such that AUB = V and ANB=0, and for every edge in E, edge e has one endpoint in A and one endpoint in B. The graph G1 given above is bipartite. 1. For each of the following four graphs, determine whether it is bipartite or not. Justify your answer. 2. Show that if a graph G = (V, E) is bipartite, then it has at most n? edges if n is even, and at most |(n2 – 1) edges if n is odd. Here n = |V| is the number of vertices in G.
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