An Exact Dominant Degree Condition for Transitive Tournament Factors in Digraphs

Abstract

Let $r\ge2$, let $T_r$ denote the transitive tournament on $r$ vertices, and write $d_G^*(v):=\max\{d_G^+(v),d_G^-(v)\}$. We prove that if $r\mid n$ and an $n$-vertex digraph $G$ satisfies $d_G^*(x)+d_G^*(y)\ge 2(1-1/r)n-1$ for every $x\ne y \in V(G)$ with $xy \notin E(G)$, then $G$ has a $T_r$-factor, and the bound is best possible. Furthermore, by applying our main theorem, we settle Treglown's conjecture on the dominant degree $d^*_G(x) \ge (1-1/r)n$ and answer Molla and Treglown's problem of determining the exact Ore-type threshold $2(1-1/r)n - 1$, and we obtain stronger versions of the theorems of Czygrinow, DeBiasio, Kierstead and Molla.

Disclosure

“+ o(1) n 2 2r · reg(r) contains a Tr -factor? Where reg(r) denote the order of the largest Tr -free regular tournament. Acknowledgment. During the preparation of this work, the authors used ChatGPT 5.6 to help identify the theorem of Kierstead and Kostochka used as a key ingredient in the proof. The authors take full responsibility for all content of this work. References [1] K. Corrádi a”

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2026-08-23T05:08:47.339576Z

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