DP-Coloring of Graphs from Random Covers

TL;DR

This paper investigates the probabilistic behavior of DP-coloring in graphs from random covers, establishing threshold phenomena related to graph density and degeneracy, and extending results to fractional DP-coloring.

Contribution

It introduces the concept of random DP-covers and analyzes the threshold behavior of DP-colorability in dense and sparse graphs, including fractional variants.

Findings

High probability of non-DP-colorability when k is below the threshold

High probability of DP-colorability when k exceeds the threshold

Sharp threshold behavior for dense graphs

Abstract

DP-coloring (also called correspondence coloring) of graphs is a generalization of list coloring that has been widely studied since its introduction by Dvo\v{r}\'{a}k and Postle in $2015$. Intuitively, DP-coloring generalizes list coloring by allowing the colors that are identified as the same to vary from edge to edge. Formally, DP-coloring of a graph $G$ is equivalent to an independent transversal in an auxiliary structure called a DP-cover of $G$. In this paper, we introduce the notion of random DP-covers and study the behavior of DP-coloring from such random covers. We prove a series of results about the probability that a graph is or is not DP-colorable from a random cover. These results support the following threshold behavior on random $k$-fold DP-covers as $\rho\to\infty$ where $\rho$ is the maximum density of a graph: graphs are non-DP-colorable with high probability when $k$ is sufficiently smaller than $\rho/\ln\rho$, and graphs are DP-colorable with high probability when $k$ is sufficiently larger than $\rho/\ln\rho$. Our results depend on $\rho$ growing fast enough and imply a sharp threshold for dense enough graphs. For sparser graphs, we analyze DP-colorability in terms of degeneracy. We also prove fractional DP-coloring analogs to these results.