Question

I saw this question on a forum: http://www.geeksforgeeks.org/archives/19042

Given an undirected graph and a number m, determine if the graph can be colored with at most m colors such that no two adjacent vertices of the graph are colored with same color.

I am wondering if you can just compare the number of vertices to that of m,
instead of trying to find a particular solution?

What am I missing?

Answer 1

There could be a coloring even if the number of vertices (|V|) is bigger then m.

For example, in bipartite graph – there is coloring for any m>=2, regardless of the number of vertices.

In a clique however, the only feasible colorings require m >= |V|

So:

I am wondering if you can just compare the number of vertices to that
of m, instead of trying to find a particular solution?

If m > = |V| – there is a solution, however, if m < |V| – we can derive nothing. There might be an answer anyway.

Bonus: The graph coloring, for the general case is one of the classical NP-Complete problems – meaning – there is no known polynomial solution for it, and if one can be found – we can derive P = NP