(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G. (a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...
6. Let si = 4 and sn +1 (sn +-) for n > 0. Prove lim n→oo sn exists and find limn-oo Sn. (Hint: First use induction to show sn 2 2 and the.show (sn) is decreasing)
2. (a) Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity K(G) of G satisfies d-1 (b) A connected graph is called unicyclic if it contains exactly one cycle. Prove that the edge-connectivity of any unicyclic graph is at most 2. 2. (a) Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity K(G) of G satisfies d-1 (b) A connected...
Prove that if G is a connected graph of order n ≥ 2, then the vertices of G can be listed as v1, v2, . . . , vn such that each vertex vi (2 ≤ i ≤ n) is adjacent to some vertex in the set {v1, v2, . . . , vi−1}.
Problem 1. Consider a graph G with n vertices: • Ifq is the size of the largest independent set in graph G, show that ax(G) n. • Use the previous result to prove that x(G)(n - d) > n and thuse x(G) > n/(n - d), where d is the minimal degree of the vertex in G. • Prove that x(G) + X(G) Sn +1 (Hint: Induction). • x(G)x(G) 2 n. • Use previous inequality to show that x(G) +...
Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity κ(G) of G satisfies d-1 Let G be a connected non-complete graph with order n 2 3 and diameter d. Prove that the connectivity κ(G) of G satisfies d-1
7n where x(G)is the cromatic number of G -2n Using extremal graph theory , prove thot for any G graph x(G 7n where x(G)is the cromatic number of G -2n Using extremal graph theory , prove thot for any G graph x(G
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...
5. Let G be a graph with order n and size m. Suppose that n 2 3 and n-n2)+2 m > Using Ore's Theorem, prove that G is Hamiltonian 5. Let G be a graph with order n and size m. Suppose that n 2 3 and n-n2)+2 m > Using Ore's Theorem, prove that G is Hamiltonian
(g) Prove that Rn - {0} and sn-1 X R are homeomorphic spaces. (Hint: Consider the function f: sn-1 X R → RN- {0} given by f(x,t) = 2'x.)