3.3 2911279 Give an infinite family of graphs G for which bm_Y,(G)=2p/3.
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- Consider a connected, undirected graph G with n vertices and m edges. The graph G has a unique cycle of length k (3 <= k <= n). Prove that the graph G must contain at least k vertices of degree 2.Prove or disprove that a graph is bipartite if and only if every induced cycle has even length.Given an undirected graph G = (V.E), determine whether G contains a cycle of oddlength as subgraph.
- Prove that in each component of the context of any graph, the number of vertices of an odd degree is always even.Consider a graph G that has k vertices and k −2 connected components,for k ≥ 4. What is the maximum possible number of edges in G? Proveyour answer.You are given a simple graph. Prove that there must be at least two vertices with the same degree.
- Let G be a connected graph that has exactly 4 vertices of odd degree: v1,v2,v3 and v4. Show that there are paths with no repeated edges from v1 to v2, and from v3 to v4, such that every edge in G is in exactly one of these paths.a) List all of the paths from 1 to 4 in the graph S with a length of 5. b) The adjacent matrix for the graph R U SLook for a method that, given a linked network with even-degree vertices, can generate an Eulerian cycle in O(n+e) time.
- . Prove the following.(Note: Provide each an illustration for verification of results)Let H be a spanning subgraph of a graph G.i. If H is Eulerian, then G is Eulerian.ii. If H is Hamiltonian, then G is HamiltonianFor any graph G and H ⊆ G, we have G[V (G − E(H))] = G.True or False?Draw a graph with four vertices of degrees 1, 2, 2, and 7 or else explain whyno such graph exists.