Cyclic, diagonal and facial colorings

  • Authors:

  • Daniel Král;Tomáš Madaras;Riste Škrekovski

  • Affiliations:

  • Institute for Theoretical Computer Science, Charles University, Malostranské náměstí 25, 118 00 Prague, Czech Republic;Institute of Mathematics, P.J. Šafárik University, Jesenná 5, 04154 Košice, Slovakia;Institute for Theoretical Computer Science, Charles University, Malostranské náměstí 25, 118 00 Prague, Czech Republic and Department of Mathematics, University of Ljubljana, J ...

  • Venue:
  • European Journal of Combinatorics - Special issue: Topological graph theory II
  • Year:
  • 2005

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Abstract

In an l-facial coloring, any two different vertices that lie on the same face and are at distance at most l on that face receive distinct colors. The concept of facial colorings extends the well-known concept of cyclic colorings. We prove that ⌈18l/5⌉ + 2 colors suffice for an l-facial coloring of a plane graph. For l = 2, 3 and 4, the upper bounds of 8, 12 and 15 colors are shown. We conjecture that each plane graph has an l-facial coloring with at most 3l + 1 colors. Our results on facial colorings are used to decrease to 16 the upper bound on the number of colors needed for 1-diagonal colorings of plane quadrangulations.