Extending partial 5-colorings and 6-colorings in planar graphs

  • Authors:

  • Daniel Král

  • Affiliations:

  • Department of Applied Mathematics, Institute for Theoretical Computer Science (ITI)2, Faculty of Mathematics and Physics, Charles University, Malostranské nám., Prague, Czech Republic

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2006

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Abstract

Let D be a disc and let X be a finite set of points on the boundary of D. A set C of k-colorings of the points X is called k-feasible if there exists a plane graph G drawn in the disc D with X ⊆ V(G) such that precisely colorings contained in the set C can be extended to proper k-colorings of G. We show that for each k-feasible set C, there exists such a plane graph G of order at most |X| + 5|X| if k = 5 and 17|X| - 48 if k = 6.