## Abstract

A list assignment L of G is a mapping that assigns every vertex vâ̂̂V(G) a set L(v) of positive integers. For a given list assignment L of G, an (L,r)-coloring of G is a proper coloring c such that for any vertex v with degree d(v), c(v)â̂̂L(v) and v is adjacent to at least min{d(v),r} different colors. The r-hued chromatic number of G, ^{χr}(G), is the least integer k such that for any vâ̂̂V(G) with L(v)={1,2,.,k}, G has an (L,r)-coloring. The r-hued list chromatic number of G, χL,_{r}(G), is the least integer k such that for any vâ̂̂V(G) and every list assignment L with |L(v)|=k, G has an (L,r)-coloring. Let K(r)=r+3 if 2≤r≤3, and K(r)=âŒŠ3r/2âŒ+1 if r≥4. We proved that if G is a ^{K4}-minor free graph, then^{χr}(G)≤K(r), and the bound can be attained;χL,_{r}(G)≤K(r)+1. This extends a former result in Lih et al. (2003).

Original language | English (US) |
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Pages (from-to) | 47-52 |

Number of pages | 6 |

Journal | Discrete Mathematics |

Volume | 315-316 |

Issue number | 1 |

DOIs | |

State | Published - 2014 |

Externally published | Yes |

## Keywords

- (k, r)-coloring
- Planar graph
- r-hued coloring
- r-hued list coloring

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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