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A star-comb lemma for infinite digraphs

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arxiv 2406.04877 v3 pith:ZDDPU6M6 submitted 2024-06-07 math.CO

classification math.CO
keywords infiniteconnectedgraphlemmastar-combcombdirectedeither
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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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  1. Halin's grid theorem for digraphs

    math.CO 2024-12 conditional novelty 7.0 of 10

    Every infinite family of disjoint equivalent directed rays in a digraph contains a directed quarter-grid with those rays as vertical rays, with an analogous result for necklace-based ends.

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