Counting List Colorings of Unlabeled Graphs
Hemanshu Kaul, Jeffrey A. Mudrock

TL;DR
This paper extends the concept of counting graph colorings to unlabeled graphs by defining an unlabeled list color function and investigates when it equals the traditional chromatic polynomial for large k.
Contribution
It introduces the unlabeled list color function for graphs and proves its equality with the chromatic polynomial for a broad class of graphs when k is sufficiently large.
Findings
The unlabeled list color function is defined as an extension of Hanlon's chromatic polynomial.
For many graphs, including twin-free graphs, the list color function matches the chromatic polynomial for large k.
Abstract
The classic enumerative functions for counting colorings of a graph , such as the chromatic polynomial , do so under the assumption that the given graph is labeled. In 1985, Hanlon defined and studied the chromatic polynomial for an unlabeled graph , . Determining amounts to counting colorings under the action of automorphisms of . In this paper, we consider the problem of counting list colorings of unlabeled graphs. We extend Hanlon's definition to the list context and define the unlabeled list color function, , of an unlabeled graph . In this context, we pursue a fundamental question whose analogues have driven much of the research on counting list colorings and its generalizations: For a given unlabeled graph , does …
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems
