An inequality for the chromatic number of a graph

Abstract

where k = k(G) is the chromatic number of G, with equality if and only if G is a complete graph or an odd circuit. More recently Wilf [2] sharpened Brooks' inequality to k(G) ~< 1 + ~, (2) where A = A(G) is the largest eigenvalue of the vertex-adjacency matrix of G, i.e., the matrix whose i, j entry is 1 is vertices i and j are connected and 0 otherwise. In the case of a star graph with n vertices (k = 2), (1) gives only k ~ n whereas (2) gives 0(V'n). In examination it appears that the only properties of the function A(G) needed in the proof of (2) are P~. G' C G ~ A(G') ~ ,~(G).