Year: 2026
Type: preprint
Abstract: We construct near-spanning balanced antidirected trees that fail to embed in highly symmetric tournaments despite the conjectured semidegree condition being satisfied. More precisely, for every sufficiently large odd integer $n$, we exhibit a regular vertex-transitive tournament on $n$ vertices and a balanced antidirected caterpillar on $n-1$ vertices with maximum degree at most $(1+o(1))n/\log_2 n$ that is not contained in the tournament. The host satisfies the strict $k/2$ threshold in both the semidegree and pseudo-semidegree settings, where $k$ is the number of arcs of the target tree. Consequently, this gives counterexamples to the uniform sublinear-degree interpretations of Conjectures~6.8 and~7.6 in Stein's survey (2024). The order $n-1$ of the target is best possible under the strict semidegree hypothesis. We also give a six-vertex balanced antidirected double-star missing from the seven-vertex Paley tournament, and show that seven is the smallest host order for a pseudo-semidegree counterexample with this fixed double-star.
Source: arXiv (Cornell University)
Authors: Gutin, Gregory, Hao, Yiming
Institutions: Royal Holloway University of London, Nankai University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7220403633