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Bipartite graphs Consider the following graph G1 edge set of G1 is E = {(v1, v5), (V1, V6), (v2, v3), (v2, v7), (U3, 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 Let A = {v1, v3, v4, v7} and B = {v2, v5, v6}. Observe that (1) A C V and BCV are subsets of V, (2) their union AUB = 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 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 CV 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 |(n2 – 1) edges if n is odd. Here n = |V| is the number of vertices in G. (V, E) is bipartite, then it has at most n2 edges if n is even, and at most

Bipartite graphs Consider the following graph G1 edge set of G1 is E = {(v1, v5), (V1, V6), (v2, v3), (v2, v7), (U3, 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 Let A = {v1, v3, v4, v7} and B = {v2, v5, v6}. Observe that (1) A C V and BCV are subsets of V, (2) their union AUB = 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 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 CV 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 |(n2 – 1) edges if n is odd. Here n = |V| is the number of vertices in G. (V, E) is bipartite, then it has at most n2 edges if n is even, and at most

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Bipartite graphs.

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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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