Planar graphs without cycles of length 4 or 5 are $(7m:2m)$-DP-colorable
Xiaoyan Xu, Xuding Zhu
TL;DR
This work resolves a DP-coloring analogue for planar graphs with forbidden cycles of lengths 4 and 5 by combining a boundary-cycle extension approach with a discharging proof. The authors show that for every positive integer m, any graph G in P_{4,5} is $(7m:2m)$-DP-colorable, hence $(7m,2m)$-choosable, improving prior multi-coloring bounds. A key technical device is extending colorings from a boundary cycle of length at most 7, together with DP-coloring results for small trees used as reducible configurations. The analysis yields a bound on the strong fractional DP-chromatic number for this graph family and advances understanding of coloring under correspondence coloring in planar graphs with cycle-length restrictions.
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, Král', 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 Dvǒrák 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.