分数二色数与锦标赛中的支配数

中文摘要 AI 辅助

Bourneuf、Charbit和Thomassé [BCT25] 证明了锦标赛的支配数可以表示为其分数二色数的函数。所证明的函数是指数级的，且所用工具基于VC维。在本文中，我们给出了该定理的两个新证明。第一个证明基于将问题归约为对$(1/2-\epsilon)$-多数锦标赛的支配数进行界定，而[BCT25]以及Charikar、Ramakrishnan和Wang [CRW26]给出了该问题的紧界。该证明得到的支配数指数界与[BCT25]相同。第二个证明给出了支配数关于分数二色数的拟线性界。该证明通过人工智能获得，并受到Ramakrishnan [Ram26]最近关于大小为五的Condorcet获胜集合存在性的书籍证明的启发。

英文摘要

Bourneuf, Charbit and Thomassé [BCT25] showed that the domination number of a tournament can be bounded as a function of its fractional dichromatic number. The function proved was exponential and the tools were based on VC-dimension. In this paper, we present two new proofs of this theorem. The first proof is based on a reduction to the problem of bounding the domination number of a $(1/2-ε)$-majority tournament, for which [BCT25] and Charikar, Ramakrishnan and Wang [CRW26] gave tight bounds. This proof yields the same exponential bound on the domination number as in [BCT25]. The second proof gives a quasilinear bound for the domination in terms of the fractional dichromatic number. It was obtained via AI and was inspired by the recent book proof of the existence of a Condorcet Winning Set of size five due to Ramakrishnan [Ram26].

发表机构

• G-SCOP, Université Grenoble Alpes（格勒诺布尔阿尔卑斯大学）
• LIP, CNRS, ENS de Lyon（里昂高等师范学院）

机构由 AI 辅助整理，请以论文原文为准。