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

Author

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

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.

Suggested Citation

  • Chen, Ye & Fan, Suohai & Lai, Hong-Jian & Song, Huimin & Xu, Murong, 2020. "Decomposition and r-hued Coloring of K4(7)-minor free graphs," Applied Mathematics and Computation, Elsevier, vol. 384(C).
    • Handle: RePEc:eee:apmaco:v:384:y:2020:i:c:s0096300320301752
    DOI: 10.1016/j.amc.2020.125206
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    Cited by:

    1. Liang, Lingmei & Liu, Fengxia & Lai, Hong-Jian, 2023. "On list r-hued coloring of outer-1-planar graphs," Applied Mathematics and Computation, Elsevier, vol. 440(C).

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