Coloring locally bipartite graphs on surfaces

  • Authors:

  • Bojan Mohar;Paul D. Seymour

  • Affiliations:

  • Department of Mathematics, University of Ljubljana, 1111 Ljubljana, Slovenia;Department of Mathematics, Princeton University, Princeton, New Jersey

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2002

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Abstract

It is proved that there is a function f: N → N such that the following holds. Let G be a graph embedded in a surface of Euler genus g with all faces of even size and with edge-width ≥ f(g). Then (i) If every contractible 4-cycle of G is facial and there is a face of size 4, then G is 3-colorable. (ii) If G is a quadrangulation, then G is not 3-colorable if and only if there exist disjoint surface separating cycles C1, ..., Cg such that, after cutting along C1, ..., Cg, we obtain a sphere with g holes and g Möbius strips, an odd number of which is nonbipartite.If embeddings of graphs are represented combinatorially by rotation systems and signatures [5], then the condition in (ii) is satisfied if and only if the geometric dual of G has an odd number of edges with negative signature.