
Let n >k> 1 with n even and k odd. Make a k-regular graph G by putting n vertices in a circle and...
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
3. Let G be an undirected graph in which the degree of every vertex is at least k. Show that there exist two vertices s and t with at least k edge-disjoint paths between them.
Let G = (V;E) be an undirected and unweighted graph. Let S be a subset of the vertices. The graph induced on S, denoted G[S] is a graph that has vertex set S and an edge between two vertices u, v that is an element of S provided that {u,v} is an edge of G. A subset K of V is called a killer set of G if the deletion of K kills all the edges of G, that is...
Graph theory
a) prove that there is a 3 regular graph if n is even and
n>=4
b)
thank you
Exercise 4. How many distinct graphs are there with vertex set {1,2,...,n}? How many distinct graphs are there with vertex set {1,2,...,n} and m edges? For these questions, what happens if "with vertex set {1,2,...,n}" is replaced by "with n vertices"?
Question 13. Prove that if k is odd and G is a k-regular (k - 1)-edge-connected graph, then G has a perfect matching
Question 13. Prove that if k is odd and G is a k-regular (k - 1)-edge-connected graph, then G has a perfect matching
Prove that if u and v are two vertices of a k-critical graph G, then N(u) 6⊆ N(v).
Let G be a graph with n vertices and n edges. (a) Show that G has a cycle. (b) Use part (a) to prove that if G has n vertices, k components, and n − k + 1 edges, then G has a cycle.
Problem 8. (2+4+4 points each) A bipartite graph G = (V. E) is a graph whose vertices can be partitioned into two (disjoint) sets V1 and V2, such that every edge joins a vertex in V1 with a vertex in V2. This means no edges are within V1 or V2 (or symbolically: Vu, v E V1. {u, u} &E and Vu, v E V2.{u,v} &E). 8(a) Show that the complete graph K, is a bipartite graph. 8(b) Prove that no...
7.5 (i) Prove that, if G is a bipartite graph with an odd number of vertices, then G is non-Hamiltonian. (ii) Deduce that the graph in Fig. 7.7 is non-Hamiltonian. Fig. 7.7 (iii) Show that, if n is odd, it is not possible for a knight to visit all the squares of an n chessboard exactly once by knight's moves and return to its starting point.
Let G be a graph in which there is a cycle C odd length that has vertices on all of the other odd cycles. Prove that the chromatic number of G is less than or equal to 5.
PLEASE HELP Let G is a graph with 2n vertices and n^2 edges. An amicable pair of vertices is an unordered pair (u, v), such that dist(u, v) = 2. Prove that G has at least n(n − 1) amicable pairs of vertices.