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)rk(G) and Rk(G)Rk(G), respectively. van Nuffelen conjectured that for any graph G  , χ(G)⩽rk(G)χ(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  , χ(G)⩽Rk(G)χ(G)⩽Rk(G). Here we improve this upper bound and show that χl(G)⩽(rk(G)+Rk(G))/2χl(G)⩽(rk(G)+Rk(G))/2, where χl(G)χl(G) is the list chromatic number of G.