Coloring Graphs having Few Colorings over Path Decompositions
Andreas Bj\"orklund

TL;DR
This paper introduces a new Monte Carlo algorithm for graph coloring that is efficient for graphs with few colorings and high maximum degree, leveraging a novel variation of the Alon–Tarsi theorem to distinguish colorable graphs.
Contribution
The authors develop a faster algorithm for coloring graphs with few colorings when the number of colors exceeds half the maximum degree, using a new algebraic approach based on a modified Alon–Tarsi theorem.
Findings
Algorithm distinguishes between k-colorable graphs with few colorings and non-k-colorable graphs.
Runs in loor(/2)old pathwidth-based time, improving previous bounds.
Avoids SETH-based hardness by focusing on graphs with high-degree vertices.
Abstract
Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no time algorithm for deciding if an -vertex graph with pathwidth admits a proper vertex coloring with colors unless the Strong Exponential Time Hypothesis (SETH) is false. We show here that nevertheless, when , where is the maximum degree in the graph , there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph along with a path decomposition of with pathwidth runs in time, that distinguishes between -colorable graphs having at most proper -colorings and non--colorable graphs. We also show how to obtain a…
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Taxonomy
TopicsAdvanced Graph Theory Research · Complexity and Algorithms in Graphs · Limits and Structures in Graph Theory
