Let G be a connected graph. Prove that G is Eulerian if and only if each of the blocks of G is Eulerian.
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- Let G be a simple connected graph with n vertices and 1/2(n-1)(n-2)+2 edges. Use Ore's theorem to prove that G is Hamiltonian.Prove that if v0 and v1 are distinct vertices of a graph G = (V,E) and a path exists in G from v0 to v1 , then there is a simple path in G from v0 to v1 .Let G be a simple graph with 11 vertices, each of degree 5 or 6. Prove that G has at least 7 vertices of degree 8 or at least 6 vertices of degree 7. Do not use the planar equation e <= 3v - 6.
- Let G be a graph. Prove that G is Eulerian if and only if G has an orientation D where D is an Eulerian digraph.The symmetric difference graph of two graphs G1 = (V, E1) and G2 = (V, E2) on the samevertex set is defined as G1△G2 := (V, E1△E2). Remember that E1△E2 := (E1 \ E2) ∪(E2 \ E1). If G1 and G2 are eulerian, show that every vertex in G1△G2 has even degreeLet G be a connected graph of order n and size n. Prove that G contains a single cycle.
- Let G1 = (A1, B1) and G2 = (A2, B2) be interval-valued fuzzy strong graphs. Then G1 is isomorphic to G2 if and only if the complement of G1is isomorphic to the complement of G2.Let G be a graph. Prove that if G has a closed walk of odd length, that the vertices and edges of the walk contain a cycle.Let G be a bipartite Prove that a(G) = graph of order n. if and only if G has a perfect matching.
- Prove that graphs G1 and G2 are isomorphic.Let u and v be distinct vertices in a connected graph G. There may be several connected subgraphs of G containing u and v. What is the minimum size of a connected subgraph of G containing u and v? Explain your answer.Let G = (V, E) be a connected graph with a bridge e = uv. Prove that there exist two disjoint sets of vertices U,W whose union is V where any path of G from vertices of U to vertices of W contains e.