Multiple DP-coloring of planar graphs without 3-cycles and normally adjacent 4-cycles

TL;DR

This paper proves that certain planar graphs without specific small cycles are colorable under a generalized DP-coloring scheme, establishing bounds on their fractional choice number.

Contribution

It introduces a new DP-coloring result for planar graphs without 3-cycles and adjacent 4-cycles, extending previous coloring theories.

Findings

Planar graphs without 3-cycles and adjacent 4-cycles are $(7m, 2m)$-DP-colorable.

The strong fractional choice number of these graphs is at most 7/2.

The result generalizes existing coloring bounds for specific planar graph classes.

Abstract

The concept of DP-coloring of a graph is a generalization of list coloring introduced by Dvo\v{r}\'{a}k and Postle in 2015. Multiple DP-coloring of graphs, as a generalization of multiple list coloring, was first studied by Bernshteyn, Kostochka and Zhu in 2019. This paper proves that planar graphs without 3-cycles and normally adjacent 4-cycles are $(7m, 2m)$-DP-colorable for every integer $m$. As a consequence, the strong fractional choice number of any planar graph without 3-cycles and normally adjacent 4-cycles is at most $7/2$.