A vertex coloring of a graph is an assignment of natural number labels called co

Question

A vertex coloring of a graph is an assignment of natural number labels called colors to each vertex. A valid vertex coloring of a graph is a vertex coloring such that no pair of adjacent vertices share the same color. Consider the following algorithm for greedily coloring the vertices of a graph: for each vertex in the graph (taken in a random order), set that vertex’s color to be the lowest natural number that is not shared by any adjacent vertex. Give a brief argument explaining why this algorithm results in a valid coloring. What is the maximum number of colors the algorithm might use?

Explanation / Answer

we are afraid there is no efficient algorithm available for coloring a graph with minimum number of colors. Greedy algorithm does not guaramtees the use of minium number of colors, but it guarantees an upper bound on this number of colors. The basic greedy coloring algorithm never uses more than d+1 colors where d is the maximum degree of vertices.

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