Decomposition and r-hued Coloring of K4(7)-minor free graphs

Ye Chen, Suohai Fan, Hong Jian Lai, Huimin Song, Murong Xu

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

A (k, r)-coloring of a graph G is a proper k-vertex coloring of G such that the neighbors of each vertex of degree d will receive at least min{d, r} different colors. The r-hued chromatic number, denoted by χr(G), is the smallest integer k for which a graph G has a (k, r)-coloring. Let f(r)=r+3 if 1 ≤ r ≤ 2, f(r)=r+5 if 3 ≤ r ≤ 7 and f(r)=⌊3r/2⌋+1 if r ≥ 8. In [Discrete Math., 315-316 (2014) 47-52], an extended conjecture of Wegner is proposed that if G is planar, then χr(G) ≤ f(r); and this conjecture was verified for K4-minor free graphs. For an integer n ≥ 4, let K4(n) be the set of all subdivisions of K4 on n vertices. We obtain decompositions of K4(n)-minor free graphs with n ∈ {5, 6, 7}. The decompositions are applied to show that if G is a K4(7)-minor free graph, then χr(G) ≤ f(r) if and only if G is not isomorphic to K6.

Original languageEnglish (US)
Article number125206
JournalApplied Mathematics and Computation
Volume384
DOIs
StatePublished - Nov 1 2020

Keywords

  • (k, r)-coloring
  • Coloring
  • Decompositions
  • Graph minor
  • r-hued list coloring

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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