let G be an n vertex simple graph other than Kn Prove that if G is not k-connected, than G has a separating set of size k-1
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- Let G be a simple graph with 15 vertices and 4 connected components. Prove thatG has at least one component with at least 4 vertices. What is the largest numberof vertices that a component of G have? Compute the maximum number of edgesof GLet G be a graph with n ≥ 3 vertices that has a clique of size n − 2 but no cliques of size n − 1.Prove that G has two distinct independent sets of size 2. Show your work and complete proof.Let G be a graph with n vertices and an independent set of size s ≥ 1. What is the maximum possible number of edges in G? Show that this bound is sharp.
- Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex. Let G be a graph of order n and size strictly less than n - - 1. Prove that G is not connected.Let G be a simple undirected graph (without multiple edges and loops) with n vertices and m edges. What is the maximal m?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.
- Use the Handshaking Lemma to show that a graph G always has an even number of vertices with odd degrees.A graph is bipartite if its vertex set can be partitioned into two sets V₁ and V2 such all edges are between V₁ and V2 (i.e. there are no edges joining vertices inside V₁, and the same for V2). (a) Draw a bipartite graph with 5 vertices and 5 edges. (b) What is the maximum number of edges for a bipartite graph with 2n vertices (suppose n > 1)?Let G be a connected planar graph of order n > 3 and has no cycles of length 3. Prove that q≤2n4, where q is the number of edges in G.
- Let G be a simple graph with exactly 11 vertices. Prove that G or its complement G must benon-planar. Hint: The maximum number of edges in a planar graph with n vertices is 3n − 6.Please write in complete sentences, include all details, showall of your work, and clarify all of your reasoning.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.Give an upper bound on the number e of edges of G in terms of n and g if G is a connected plane graph with n vertices and girth g.