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.

Keywords

  1. chromatic number
  2. crossing number
  3. clique number
  4. girth

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Published In

cover image SIAM Journal on Discrete Mathematics
SIAM Journal on Discrete Mathematics
Pages: 401 - 422
ISSN (online): 1095-7146

History

Submitted: 27 January 2010
Accepted: 29 January 2011
Published online: 24 March 2011

Keywords

  1. chromatic number
  2. crossing number
  3. clique number
  4. girth

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