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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 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 is sufficiently smaller than , and graphs are DP‐colorable with high‐probability when is sufficiently larger than . Our results depend on 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.
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Efficiently list‐edge coloring multigraphs asymptotically optimally
Abstract We give polynomial time algorithms for the seminal results of Kahn, who showed that the Goldberg–Seymour and list‐coloring conjectures for (list‐)edge coloring multigraphs hold asymptotically. Kahn's arguments are based on the probabilistic method and are non‐constructive. Our key insight is that we can combine sophisticated techniques due to Achlioptas, Iliopoulos, and Kolmogorov for the analysis of local search algorithms with correlation decay properties of the probability spaces on matchings used by Kahn in order to construct efficient edge‐coloring algorithms.
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- Award ID(s):
- 1815328
- PAR ID:
- 10444388
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 61
- Issue:
- 4
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 724-753
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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