Iterated Arc Graphs
Danny Rorabaugh, Claude Tardif, David Wehlau, Imed Zaguia

TL;DR
This paper investigates the chromatic number of iterated arc graphs, showing that for symmetric digraphs, the chromatic number of the iterated arc graph depends solely on the original graph's chromatic number.
Contribution
It extends the understanding of arc graphs by demonstrating that the chromatic number of iterated arc graphs for symmetric digraphs is determined entirely by the original graph's chromatic number.
Findings
Chromatic number of $ ext{δ}^k(G)$ depends only on $G$'s chromatic number for symmetric graphs.
Generalization of Poljak and R"{o}dl's result to iterated arc graphs.
Distinct chromatic numbers can occur for non-symmetric graphs with the same original chromatic number.
Abstract
The arc graph of a digraph is the digraph with the set of arcs of as vertex-set, where the arcs of join consecutive arcs of . In 1981, Poljak and R\"{o}dl characterised the chromatic number of in terms of the chromatic number of when is symmetric (i.e., undirected). In contrast, directed graphs with equal chromatic numbers can have arc graphs with distinct chromatic numbers. Even though the arc graph of a symmetric graph is not symmetric, we show that the chromatic number of the iterated arc graph still only depends on the chromatic number of when is symmetric.
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Taxonomy
TopicsAdvanced Graph Theory Research · Limits and Structures in Graph Theory · Topological and Geometric Data Analysis
