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Weak degeneracy of the square of K4-minor free graphs

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  • Ye, Jing
  • Zou, Jiani
  • Han, Miaomiao

Abstract

A graph G is called weakly f-degenerate with respect to a function f from V(G) to the non-negative integers, if every vertex of G can be successively removed through a series of valid Delete and DeleteSave operations. The weak degeneracy wd(G) is defined as the smallest integer d for which G is weakly d-degenerate, where d is a constant function. It was demonstrated that one plus the weak degeneracy can act as an upper bound for list-chromatic number and DP-chromatic number. Let κ(G2)=Δ(G)+2 if 2≤Δ(G)≤3, and κ(G2)=⌊3Δ(G)2⌋ if Δ(G)≥4. In this paper, we prove that for every K4-minor free graph G, wd(G2)≤κ(G2), which implies that G2 is (κ(G2)+1)-choosable and (κ(G2)+1)-DP-colorable. This work generalizes the result obtained by Lih et al. in [Discrete Mathematics, 269 (2003), 303-309] and Hetherington et al. in [Discrete Mathematics, 308 (2008), 4037-4043].

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  • Ye, Jing & Zou, Jiani & Han, Miaomiao, 2025. "Weak degeneracy of the square of K4-minor free graphs," Applied Mathematics and Computation, Elsevier, vol. 500(C).
    • Handle: RePEc:eee:apmaco:v:500:y:2025:i:c:s0096300325001663
    DOI: 10.1016/j.amc.2025.129439
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    Keywords

    Weak degeneracy; K4-minor free graphs; Square graphs;
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