5-Coloring Graphs with 4 Crossings

  • Authors:

  • Rok Erman;Frédéric Havet;Bernard Lidický;Ondřej Pangrác

  • Affiliations:

  • rok.erman@gmail.com;fhavet@sophia.inria.fr;bernard@kam.mff.cuni.cz and pangrac@kam.mff.cuni.cz;-

  • Venue:
  • SIAM Journal on Discrete Mathematics
  • Year:
  • 2011

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Abstract

We answer in the negative a question of Oporowski and Zhao [Discrete Math., 309 (2009), pp. 2948-2951] asking whether every graph with crossing number at most 5 and clique number at most 5 is 5-colorable. However, we show that every graph with crossing number at most 4 and clique number at most 5 is 5-colorable. We also show some colorability results on graphs that can be made planar by removing a few edges. In particular, we show that, if a graph with clique number at most 5 has three edges whose removal leaves the graph planar, then it is 5-colorable.