Further results on the lower bounds of mean color numbers

Abstract

Let G be a graph with n vertices. The mean color number of G, denoted by μ(G), is the average number of colors used in all n-colorings of G. This paper proves that μ(G) ≥ μ(Q), where Q is any 2-tree with n vertices and G is any graph whose vertex set has an ordering x1, x 2, . .. ,xn such that xi is contained in a K3 of G[Vi] for i = 3,4, . . . ,n, where Vi = {x1,x2, . . . ,xi}. This result improves two known results that μ(G) ≥ μ(On) where On is the empty graph with n vertices, and μ(G) ≥ μ(T) where T is a spanning tree of G. © 2004 Wiley Periodicals, Inc.