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Antidirected subgraphs of oriented graphs

Part of: Graph theory

Published online by Cambridge University Press:  06 March 2024

Maya Stein  and
Camila Zárate-Guerén
Maya Stein*
Affiliation:
Center for Mathematical Modeling (IRL CNRS 2807) and Department for Mathematical Engineering, University of Chile, Santiago, Chile
Camila Zárate-Guerén
Affiliation:
University of Birmingham, Birmingham, UK
*
Corresponding author: Maya Stein; Email: mstein@dim.uchile.cl
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Abstract

We show that for every $\eta \gt 0$ every sufficiently large $n$-vertex oriented graph $D$ of minimum semidegree exceeding $(1+\eta )\frac k2$ contains every balanced antidirected tree with $k$ edges and bounded maximum degree, if $k\ge \eta n$. In particular, this asymptotically confirms a conjecture of the first author for long antidirected paths and dense digraphs.

Further, we show that in the same setting, $D$ contains every $k$-edge antidirected subdivision of a sufficiently small complete graph, if the paths of the subdivision that have length $1$ or $2$ span a forest. As a special case, we can find all antidirected cycles of length at most $k$.

Finally, we address a conjecture of Addario-Berry, Havet, Linhares Sales, Reed, and Thomassé for antidirected trees in digraphs. We show that this conjecture is asymptotically true in $n$-vertex oriented graphs for all balanced antidirected trees of bounded maximum degree and of size linear in $n$.

Keywords

oriented graphtreesemidegreecyclepathsubdivisiondigraph

MSC classification

Primary: 05C20: Directed graphs (digraphs), tournaments
Secondary: 05C07: Vertex degrees 05C35: Extremal problems
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 33 , Issue 4 , July 2024 , pp. 446 - 466
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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