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A Linear-Time Algorithm for 7-Coloring 1-Planar Graphs

  • Zhi-Zhong Chen
  • Mitsuharu Kouno
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2747)

Abstract

A graph G is 1-planar if it can be embedded in the plane in such a way that each edge crosses at most one other edge. Borodin showed that 1-planar graphs are 6-colorable, but his proof only leads to a complicated polynomial (but nonlinear) time algorithm. This paper presents a linear-time algorithm for 7-coloring 1-planar graphs (that are already embedded in the plane). The main difficulty in the design of our algorithm comes from the fact that the class of 1-planar graphs is not closed under the operation of edge contraction. This difficulty is overcome by a structure lemma that may find useful in other problems on 1-planar graphs. This paper also shows that it is NP-complete to decide whether a given 1-planar graph is 4-colorable. The complexity of the problem of deciding whether a given 1-planar graph is 5-colorable is still unknown.

Keywords

Edge Contraction Critical Edge Small Vertex Distinct Neighbor Critical Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Zhi-Zhong Chen
    • 1
  • Mitsuharu Kouno
    • 1
  1. 1.Dept. of Math. Sci.Tokyo Denki Univ.SaitamaJapan

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