Packing chromatic number of cubic graphs

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Abstract

A packingk-coloring of a graph G is a partition of V(G) into sets V1,…,Vk such that for each 1≤i≤k the distance between any two distinct x,y∈Vi is at least i+1. The packing chromatic number, χp(G), of a graph G is the minimum k such that G has a packing k-coloring. Sloper showed that there are 4-regular graphs with arbitrarily large packing chromatic number. The question whether the packing chromatic number of subcubic graphs is bounded appears in several papers. We answer this question in the negative. Moreover, we show that for every fixed k and g≥2k+2, almost every n-vertex cubic graph of girth at least g has the packing chromatic number greater than k.

Original languageEnglish (US)
Pages (from-to)474-483
Number of pages10
JournalDiscrete Mathematics
Volume341
Issue number2
DOIs
StatePublished - Feb 2018

Keywords

  • Cubic graphs
  • Independent sets
  • Packing coloring

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics

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