Abstract
Let G be a graph with a nonempty edge set, we denote the rank of the adjacency matrix of G and the term rank of G, by rk ( G ) and Rk ( G ) , respectively. It was conjectured [C. van Nuffelen, Amer. Math. Monthly 83 (1976) 265–266], for any graph G, χ ( G ) ⩽ rk ( G ) . The first counterexample to this conjecture was obtained by Alon and Seymour [J. Graph Theor. 13 (1989) 523–525]. Recently, Fishkind and Kotlov [Discrete Math. 250 (2002) 253–257] have proved that for any graph G, χ ( G ) ⩽ Rk ( G ) . In this Note we improve Fishkind–Kotlov upper bound and show that χ ( G ) ⩽ rk ( G ) + Rk ( G ) 2 . To cite this article: S. Akbari, H.-R. Fanaï, C. R. Acad. Sci. Paris, Ser. I 340 (2005).
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