DP-Coloring of Graphs from Random Covers

TL;DR

This work analyzes DP-coloring from random DP-covers, revealing a density-driven threshold around $\rho(G)/\ln\rho(G)$: for large $\rho$, DP-colorability transitions from unlikely to likely as $k$ crosses this scale. It exploits first- and second-mMoment methods for dense graphs and a degeneracy-based greedy transversal framework to obtain high-probability colorability in sparser, degenerate graphs, with fractional DP-coloring analogs paralleling the integer case. The results establish both non-colorability and colorability regimes, define DP-thresholds and sharp DP-thresholds for graph sequences, and connect the random-cover model to random lifts and existing coloring theories. Collectively, the paper clarifies how density and degeneracy govern probabilistic DP-colorability and provides tools for predicting thresholds in dense and sparse regimes, including fractional variants.

Abstract

DP-coloring (also called correspondence coloring) of graphs is a generalization of list coloring that has been widely studied since its introduction by Dvořák 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 $ρ\to\infty$ where $ρ$ is the maximum density of a graph: graphs are non-DP-colorable with high probability when $k$ is sufficiently smaller than $ρ/\lnρ$, and graphs are DP-colorable with high probability when $k$ is sufficiently larger than $ρ/\lnρ$. Our results depend on $ρ$ 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.

Theorems & Definitions (30)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Conjecture 1.4
• Proposition 1.5
• Corollary 1.6
• Corollary 1.7
• Proposition 1.9
• Theorem 1.10
• proof