Planar graphs without cycles of length 4 or 5 are $(7m:2m)$-DP-colorable

TL;DR

This paper proves that planar graphs without cycles of length 4 or 5 are $(7m:2m)$-DP-colorable for any positive integer $m$, advancing understanding of their multiple coloring properties.

Contribution

It establishes that such graphs are $(7m:2m)$-DP-colorable, providing a new result in the multiple coloring theory of these planar graphs.

Findings

Graphs are $(7m:2m)$-DP-colorable for all positive integers $m$.

This result implies $(7m,2m)$-choosability of these graphs.

Advances the understanding of coloring properties of planar graphs without 4- or 5-cycles.

Abstract

It was conjectured by Steinberg in 1976 that planar graphs without cycles of length 4 or 5 are 3-colorable. This conjecture attracted a substantial amount of attention and was finally refuted by Cohen-Addad, Hebdige, Kr\'{a}l', Li and Salgado in 2017. Although Steinberg's conjecture is settled, coloring of this family of graphs, as well as some other families of planar graphs forbidding certain cycle lengths have been attracting a lot of recent attention and many challenging problems remain open. One problem of interest is multiple coloring and multiple list coloring of this family of graphs. It was proved by Dv\v{o}r\'{a}k and Hu that planar graphs without cycles of length 4 or 5 are $(11,3)$-colorable, and this result was improved by Wang, who proved that graphs in this family are $(7:2)$-colorable. On the other hand, it was proved by Xu and Zhu that for every positive integer $m$, there is a graph in this family which is not $(3m + \lfloor \frac{m-1}{12} \rfloor, m)$-choosable. In this paper, we prove that for any positive integer $m$, graphs in this family are $(7m:2m)$-DP-colorable, and hence $(7m,2m)$-choosable.