Coloring squares of planar graphs with girth six

Authors:
Zdenk Dvořák;Daniel Král';Pavel Nejedlý;Riste Škrekovski
Affiliations:
Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI)11The Institute for Theoretical Computer Science (ITI) is supported as project 1M0545 by the Ministry ofEducat ...;Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI)11The Institute for Theoretical Computer Science (ITI) is supported as project 1M0545 by the Ministry ofEducat ...;Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI)11The Institute for Theoretical Computer Science (ITI) is supported as project 1M0545 by the Ministry ofEducat ...;Department of Mathematics, University of Ljubljana, Jadranska19, 1111 Ljubljana, Slovenia
Venue:
European Journal of Combinatorics
Year:
2008

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Abstract

Wang and Lih conjectured that for every g=5, there exists a number M(g) such that the square of a planar graph G of girth at least g and maximum degree @D=M(g) is (@D+1)-colorable. The conjecture is known to be true for g=7 but false for g@?{5,6}. We show that the conjecture for g=6 is off by just one, i.e., the square of a planar graph G of girth at least six and sufficiently large maximum degree is (@D+2)-colorable.