Coloring graphs with fixed genus and girth

It is well known that the maximum chromatic number of a graph on the orientable surface Sg is 0(g1/2). We prove that there are positive constants Cl, C2 such that every triangle-free graph on Sg has chromatic number less than c2(g/ log(g))1/3 and that some triangle-free graph on Sg has chromatic number at least cl l/g3) . We obtain similar results for graphs with restricted clique number or girth on Sg or Nk. As an application, we prove that an Sg-polytope has chromatic number at most 0(g3/7). For specific surfaces we prove that every graph on the double torus and of girth at least six is 3-colorable and we characterize completely those triangle-free projective graphs that are not 3-colorable.

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