## Abstract

A packingk-coloring of a graph G is a partition of V(G) into sets V_{1},…,V_{k} such that for each 1≤i≤k the distance between any two distinct x,y∈V_{i} is at least i+1. The packing chromatic number, χ_{p}(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 language | English |
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Pages (from-to) | 474-483 |

Number of pages | 10 |

Journal | Discrete Mathematics |

Volume | 341 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2018 |

Externally published | Yes |

## Keywords

- Cubic graphs
- Independent sets
- Packing coloring