Let n ≥ 2 and p ≥ 1 be two positive integers. Let G be a graph with n vertices such that each vertex has p or more incident edges. Prove that if p > (n−2)/2, then G is connected using proof by contradiction.

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.

please explain fully

please answer both of the questions.   7. The Bellman-Ford algorithm for single-source shortest paths on a graph G(V,E) as discussed in class has a running time of O|V |3
, where |V | is the number of vertices in the given graph. However, when the graph is sparse (i.e., |E| << |V |2), then this running time can be improved to O(|V ||E|). Describe how how this can be done..   8. Let G(V,E) be an undirected graph such that each vertex has an even degree. Design an O(|V |+ |E|) time algorithm to direct the edges of G such that, for each vertex, the outdegree is equal to the indegree.    Please give proper explanation and typed answer only.

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

Most graph algorithms that take an n×n adjacency-matrix representation as input require at least time O(n^2), but there are some exceptions. Show how to determine whether a simple directed graph G contains a universal sink, that is, a vertex with in degree n − 1 and out-degree 0, in time O(n) given an n × n adjacency matrix for G. (A vertex v has indegree k if there are precisely k edges of the form (u, v), and has outdegree k if there are precisely k edges of the form (v, u).)

Given a digraph, find a bitonic shortest path from s to every other vertex (if one exists). A path is bitonic if there is an intermediate vertex v suchthat the edges on the path from s to v are strictly increasing and the edges on the pathfrom v to t are strictly decreasing. The path should be simple (no repeated vertices).

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

Let G (V, E) be a digraph in which every vertex is a source, or a sink, or both a sink and a source. (a) Prove that G has neither self-loops nor anti-parallel edges.

For any graph G, does your algorithm always use exactly χ(G) colours? If so, explain why. If not, provide a graph G for which your algorithm requires more than χ(G) colours.