Problem 2: Let G and H be the graphs below. For each graph, determine whether it...
3. (a) Let Knbe the complete bipartite graph with n vertices in each part of its bipartition, where n 21. Determine the number of perfect matchings of Kn (b) A matching M in a graph Gis ca a mazimal matching if there exists no matching M' of G such that M is a proper subset of M' Prove that, for any graph G and any edges e,f of G which are not incident with a common vertex, there exists a...
please solve without using Konig theorem Let G be a bipartite graph of order n. Prove that a(G) = if and only if G has a perfect matching.
Please answer the question and write legibly (3) Prove that for a bipartite graph G on n vertices, we have a(G)- n/2 if and only if G has a perfect matching. (Recall that α(G) is the maximum size among the independent subsets of G.) (3) Prove that for a bipartite graph G on n vertices, we have a(G)- n/2 if and only if G has a perfect matching. (Recall that α(G) is the maximum size among the independent subsets of...
2. a) Determine whether the following graphs are isomorphic or not. If so write an isomorphism, if not explain why. 1 b 2 a 6 3 f d 5 4 e Graph A Graph B. b) Is the graph A bipartite. If not, find a vertex v such that A - v bipartite? c) Does the graph A have an Eulerian circuit? If not find an edge e such that A - e has an Eulerian circuit.
Let G -(V, E) be a graph. The complementary graph G of G has vertex set V. Two vertices are adjacent in G if and only if they are not adjacent in G. (a) For each of the following graphs, describe its complementary graph: (i) Km,.ni (i) W Are the resulting graphs connected? Justify your answers. (b) Describe the graph GUG. (c) If G is a simple graph with 15 edges and G has 13 edges, how many vertices does...
each of the following graphs has an Euler circuit. If it does have an Euler Determine whether such a circuit. If it does not have an Euler circuit, explain why you can find circuit, find be 100% sure. Ca au 2 (4) Find which of the following graphs are bipartite. Redraw the bipartite graphs so that their bipartite nature is evident. V2 5 니 each of the following graphs has an Euler circuit. If it does have an Euler Determine...
G3: I can determine whether a graph has an Euler trail (or circuit), or a Hamiltonian path (or cycle), and I can clearly explain my reasoning. Answer each question in the space provided below. 1. Draw a simple graph with 7 vertices and 11 edges that has an Euler circuit. Demonstrate the Euler circuit by listing in order the vertices on it. 2. For what pairs (m, n) does the complete bipartite graph, Km,n contain a Hamiltonian cycle? Justify your...
please throughly explain each step.47.21. What does it mean for two graphs to be the same? Let G and H be graphs. We say th G is isomorphic to H provided there is a bijection f VG)-V(H) such that for all a, b e V(G) we have a~b (in G) if and only if f(a)~f (b) (in H). The function f is called an isomorphism of G to H We can think of f as renaming the vertices of G...
What does it mean for two graphs to be the same? Let G and H be graphs. We Say that G is isomorphic to H provided there is a bijection f : V(G) rightarrow V(H) such that for all a middot b epsilon V(G) we have a~b (in G) if and only if f(a) ~ f(b) (in H). The function f is called an isomorphism of G to H. We can think of f as renaming the vertices of G...
Determine whether the given pair of graphs is isomorphic, if the graphs are not isomorphic provide an argument? A) F G B) 4) Consider the following weighted graph G below