Some relations among term rank, clique number and list chromatic number of a graph

Let G be a graph with a nonempty edge set, we denote the rank of the adjacency matrix of G and term rank of G, by rk(G) and Rk(G), respectively. van Nuffelen conjectured that for any graph G, chi(G) <= rk(G). The first counterexample to this conjecture was obtained by Alon and Seymour. In 2002, Fishkind and Kotlov proved that for any graph G, chi(G) <= Rk(G). Here we improve this upper bound and show that chi(l) (G) <= (rk (G) + Rk (G))/2, where chi(l) (G) is the list chromatic number of G. (c) 2006 Elsevier B.V. All rights reserved.