A map colour theorem for the union of graphs

Abstract

In 1890 Heawood [Map colour theorem, Quart. J. Pure Appl. Math. 24 (1890) 332–338] established an upper bound for the chromatic number of a graph embedded on a surface of Euler genus g ⩾ 1 . This upper bound became known as the Heawood number H ( g ) . Almost a century later, Ringel [Map Color Theorem, Springer, New York, 1974] and Ringel and Youngs [Solution of the Heawood map-coloring problem, Proc. Nat. Acad. Sci. USA 60 (1968) 438–445] proved that the Heawood number H ( g ) is in fact the maximum chromatic number as well as the maximum clique number of graphs embedded on a surface of Euler genus g ⩾ 1 besides the Klein bottle. In this paper, we present a Heawood-type formula for the edge disjoint union of two graphs that are embedded on a given surface Σ . More precisely, we determine the number H 2 ( Σ ) such that if a graph G embedded on Σ is the edge disjoint union of two graphs G 1 and G 2 , then ω ( G 1 ) + ω ( G 2 ) ⩽ χ ( G 1 ) + χ ( G 2 ) ⩽ H 2 ( Σ ) . Similar to the results of Ringel and Ringel and Youngs, we show that this bound is sharp for all but at most one non-orientable surface Σ .