Rank, term rank, and chromatic number

Abstract

Let G be a graph with a nonempty edge set, and with rank rk(G) , term rank Rk(G) , and chromatic number χ(G). We characterize Rk(G) as being the maximum number of colors in certain proper colorings of G. In particular, we observe that χ(G)⩽ Rk(G) , with equality holding if and only if (besides isolated vertices) G is either complete or a star. For a twin-free graph G, we observe the bound Rk(G)= O( 2 rk(G) ) and we show that this bound is sharp.