New upper bounds for the chromatic number of a graph

  • Authors:

  • Ladislav Stacho

  • Affiliations:

  • Department of Informatics, Institute of Mathematics, Slovak academy of sciences, 840 00 Bratislava 4, Slovakia

  • Venue:
  • Journal of Graph Theory
  • Year:
  • 2001

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Abstract

We show that for any graph G, the chromatic number χ(G) ≤ Δ2(G) + 1, where Δ2(G) is the largest degree that a vertex ν can have subject to the condition that ν is adjacent to a vertex whose degree is at least as big as its own. Moreover, we show that the upper bound is best possible in the the following sense: If Δ2(G) ≥ 3, then to determine whether χ(G) ≤ Δ2(G) is an NP-complete problem. © 2001 John Wiley & Sons, Inc. J Graph Theory 36: 117–120, 2001 A part of the paper was prepared during a postdoc fellowship of the author at the Department for Computer Science, University of Turku, TUCS, Finland.