TY - JOUR

T1 - Packing chromatic number of cubic graphs

AU - Balogh, József

AU - Kostochka, Alexandr

AU - Liu, Xujun

N1 - Funding Information:
We thank the referees for their valuable comments. Research of first author is partially supported by NSF Grant DMS-1500121 , Arnold O. Beckman Research Award (UIUC Campus Research Board 15006 ) and by the Langan Scholar Fund (UIUC). Research of second author is supported in part by NSF grant DMS-1266016 and by grants 15-01-05867 and 16-01-00499 of the Russian Foundation for Basic Research .
Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2018/2

Y1 - 2018/2

N2 - 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.

AB - 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.

KW - Cubic graphs

KW - Independent sets

KW - Packing coloring

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U2 - 10.1016/j.disc.2017.09.014

DO - 10.1016/j.disc.2017.09.014

M3 - Article

AN - SCOPUS:85030757274

VL - 341

SP - 474

EP - 483

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 2

ER -