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, (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, (G) Rk(G). Here we improve this upper bound and show that l (G) (rk(G)+ Rk(G))/2, where l (G) is the list chromatic number of G. © 2006 Elsevier B.V. All rights reserved.