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 t...
(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...
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.
Graph theory has at least degrees and use Theorem rove that a bipartite graph t n2-n G in which each part has order n, and G 2 edges, must be hamiltonian. Hint: Examine the 5.2 2 If G is a graph of order n 2 3 such that deg() 2 n/2 for all DEV(G), then G is hamiltonian
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
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...
Let G be a simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle is formed. Hint: Check n 2 first and then use induction Let G be a simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle is formed. Hint: Check n 2 first and then use induction
Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2. Let G be a connected graph with m 2 vertices of odd degree. Prove that once is m/2.
49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is connected. b. Show that the result in (a) is best possible; that is, for each n 2 2, prove there is a graph with ("2) edges that is not connected. 49.12. Let G be a graph with n 2 2 vertices. a. Prove that if G has at least ("21) +1 edges, then G is...
e hese seqd 7. Prove the handshaking theorem. Let G- (V,E) be an undirected graph with m edges. Then 2m Evev(degv)).
Let f(n) = 5n^2. Prove that f(n) = O(n^3). Let f(n) = 7n^2. Prove that f(n) = Ω(n). Let f(n) = 3n. Prove that f(n) =ꙍ (√n). Let f(n) = 3n+2. Prove that f(n) = Θ (n). Let k > 0 and c > 0 be any positive constants. Prove that (n + k)c = O(nc). Prove that lg(n!) = O(n lg n). Let g(n) = log10(n). Prove that g(n) = Θ(lg n). (hint: ???? ? = ???? ?)???? ?...