An improved lower bound of P(G,L)−P(G,k) for k-assignments L

Abstract

Let G=(V,E) be a simple graph with n vertices and m edges, P(G,k) be the chromatic polynomial of G, and P(G,L) be the number of L-colorings of G for any k-assignment L. In this article, we show that when k≥m−1≥3, P(G,L)−P(G,k) is bounded below by ((k−m+1)kn−3+(k−m+3)c3kn−5)∑uv∈E|L(u)∖L(v)|, where c≥(m−1)(m−3)8, and in particular, if G is K3-free, then c≥(m−22)+2m−3. Consequently, P(G,L)≥P(G,k) whenever k≥m−1.