Iterated Arc Graphs

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 $\delta(G)$ of a digraph $G$ is the digraph with the set of arcs of $G$ as vertex-set, where the arcs of $\delta(G)$ join consecutive arcs of $G$. In 1981, Poljak and R\"{o}dl characterised the chromatic number of $\delta(G)$ in terms of the chromatic number of $G$ when $G$ 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 $\delta^k(G)$ still only depends on the chromatic number of $G$ when $G$ is symmetric.