A map colour theorem for the union of graphs

  • Authors:

  • Michael Stiebitz;Riste Škrekovski

  • Affiliations:

  • Technische Universität Ilmenau, Institute of Mathematics, Ilmenau, Germany;Charles University, Malostranské nám., Prague, Czech Republic and University of Ljubljana, Jadranska, Ljubljana, Slovenia

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2006

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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 H2(Σ) such that if a graph G embedded on Σ is the edge disjoint union of two graphs G1 and G2, then ω(G1)+ω(G2)≤χ(G1)+χ(G2)≤H2(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.