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arXiv · 2406.04877

A star-comb lemma for infinite digraphs

Florian Reich
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Abstract

The star-comb lemma is a standard tool in infinite graph theory, which states that for every infinite set $U$ of vertices in a connected graph $G$ there exists either a subdivided infinite star in $G$ with all leaves in $U$, or an infinite comb in $G$ with all teeth in $U$. In this paper, we elaborate a counterpart of the star-comb lemma for directed graphs. More precisely, we prove that for every infinite set $U$ of vertices in a strongly connected directed graph $D$, there exists a strongly connected butterfly minor of $D$ with infinitely many teeth in $U$ that is either shaped by a star or shaped by a comb, or is a chain of triangles.

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Florian Reich. 2024-06-07. A star-comb lemma for infinite digraphs. https://arxiv.org/abs/2406.04877

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