A Linear Lower Bound for Dominating Sets in $k$-Majority Tournaments

Abstract

A $k$-majority tournament on a finite vertex set is defined by $2k-1$ linear orders, with $u\to v$ when $u$ lies above $v$ in at least $k$ of the orders. Let $F(k)$ be the maximum, over all $k$-majority tournaments, of the size of a minimum dominating set. Alon, Brightwell, Kierstead, Kostochka, and Winkler proved that $C_1k/\log k \leq F(k) \leq C_2k\log k$ for suitable positive constants $C_1$ and $C_2$. In this paper, we prove the linear lower bound $F(k)\ge \left\lfloor\frac{k+1}{2}\right\rfloor $ for $k\ge 3$.

Disclosure

“increase in the number of orders. In view of the upper bound in (1), the principal remaining asymptotic question is whether F (k) = O(k). Declaration on the use of generative AI During the preparation of this manuscript, the authors used OpenAI’s ChatGPT (5.6 Sol) in a limited supporting role for language polishing, presentation, bibliographic checks, and exploratory mathematical discussion. Some mathematical ideas and elements of the construction arose through interaction between the auth”

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Classification

Substantial mathematical content or result generation

Multiplier

10

Verified

2026-08-23T19:05:46.122463Z

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