DP-Coloring of Graphs from Random Covers
Anton Bernshteyn, Daniel Dominik, Hemanshu Kaul, Jeffrey A. Mudrock

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 . 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 is equivalent to an independent transversal in an auxiliary structure called a DP-cover of . 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 -fold DP-covers as where is the maximum density of a graph: graphs are non-DP-colorable with high probability when …
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Taxonomy
TopicsAdvanced Graph Theory Research
