List equitable coloring of planar graphs without $4$- and $6$-cycles when $\Delta(G)=5$

Year 2024, Volume: 53 Issue: 5, 1393 - 1400, 15.10.2024

Aijun Dong

https://doi.org/10.15672/hujms.1255155

Abstract

A graph $G$ is $k$ list equitably colorable, if for any given $k$-uniform list assignment $L$, $G$ is $L$-colorable and each color appears on at most $\lceil\frac{|V(G)|}{k}\rceil$ vertices. In 2009, Li and Bu obtained that for planar graph $G$, if $\Delta(G)\geq6$ and without $4$- and $6$-cycles, then $G$ is $\Delta(G)$ list equitably colorable. In order to further prove the conjecture of list equitable coloring, in this paper, we focus on planar graph with $\Delta(G)=5$, and prove that if $G$ is a planar graph without $4$- and $6$-cycles, then $G$ is $\Delta(G)$ list equitably colorable.

Keywords

list equitable coloring, planar graph, degenerate graph

Supporting Institution

National Natural Science Foundation of China

Project Number

NSFC12001332

Thanks

This work was supported by the National Natural Science Foundation of China (Grant No. NSFC12001332). It was also supported by China Postdoctoral Science Foundation Funded Project (Grant No.2014M561909); the Nature Science Foundation of Shandong Province of China (Grant No. ZR2014AM028, ZR2017BA009)

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