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arXiv · 1610.01259

Iterated Arc Graphs

Danny Rorabaugh·Claude Tardif·David Wehlau·Imed Zaguia
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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.

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Danny Rorabaugh, Claude Tardif, David Wehlau, Imed Zaguia. 2016-10-05. Iterated Arc Graphs. https://arxiv.org/abs/1610.01259

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