Coloring graphs of various maximum degree from random lists
Carl Johan Casselgren

TL;DR
This paper investigates the probability of properly coloring graphs with bounded maximum degree using random list assignments, establishing bounds on list size for high probability of success as the graph size grows.
Contribution
It provides new asymptotic bounds on list sizes ensuring high-probability proper colorings in graphs with various maximum degrees under random list assignments.
Findings
If $\sigma(n) = ext{omega}(n^{1/k^2} \Delta^{1/k})$, the probability of an $L$-coloring tends to 1.
For $\Delta= ext{omega}(n^{1/2})$, the same holds if $\sigma = ext{omega}(\Delta)$.
Results extend to different bounds on $\Delta$ for fixed or increasing $k$.
Abstract
Let be a graph on vertices with maximum degree . Assign to each vertex of a list of colors by choosing each list independently and uniformly at random from all -subsets of a color set of size . Such a list assignment is called a \emph{random -list assignment}. In this paper, we are interested in determining the asymptotic probability (as ) of the existence of a proper coloring of , such that for every vertex of , a so-called -coloring. We give various lower bounds on , in terms of , and , which ensures that with probability tending to 1 as there is an -coloring of . In particular, we show, for all fixed and growing , that if and…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research
