Let G be a graph with n 2 3 vertices that has a clique of size n-2 but no cliques of sizen-1. Prove that G has two distinct independent sets of size 2.

Let G be a graph with n vertices. If the maximum size of an independent set in G is k, clearly explain why the minimum size of a vertex cover in G is n - k.

Let G be a graph with n vertices. The k-coloring problem is to decide whether the vertices of G can be labeled from the set {1, 2, ..., k} such that for every edge (v,w) in the graph, the labels of v and w are different.  Is the (n-4)-coloring problem in P or in NP? Give a formal proof for your answer.  A 'Yes' or 'No' answer is not sufficient to get a non-zero mark on this question.

Consider an undirected graph G with 100 nodes. The maximum number of edges to be included in G so that the graph is not connected is

Let G = (V, E) be a connected graph that has two distinct spanning trees. Prove that |E| > |V] – 1.

Prove that in any simple graph G (i.e. no multiple edges, no loops) there exist two distinct vertices with the same degrees.

A graph is biconnected if every pair of vertices is connectedby two disjoint paths. An articulation point in a connected graph is a vertex that woulddisconnect the graph if it (and its adjacent edges) were removed. Prove that any graphwith no articulation points is biconnected. Hint : Given a pair of vertices s and t and apath connecting them, use the fact that none of the vertices on the path are articulationpoints to construct two disjoint paths connecting s and t.

Let G be a 4-regular plane graph with 10 faces. Determine how many vertices and edges G hasand give a plane drawing of such a graph. Please write complete sentences, include all details, show all of your work, and clarify all of your reasoning.

QUESTION 16 Let G be a connected graph having the following adjacency matrix: How many vertices have an odd degree? OO 23 OO 40 0 10 100 0101 31101

Let H be a planar graph with n vertices and m edges, where n ≥ 5. Prove that if H does not have any cycles of length 3 and 4 then 3m ≤ 5n - 10.