AI 中文总结
本文针对Nguyen等提出的交叉锦标赛类，改编现有方法证明其为多项式$\boldsymbol{\u03c7}$有界，并指出该结果无法推广到回边图为弦图的锦标赛。
AI 中文摘要
给定锦标赛$T$，Aboulker、Aubian、Charbit和Lopes（2023）将其团数$\boldsymbol{\u03c9}(T)$定义为$T$的回边图的最小团数，并提出问题：哪些锦标赛类是多项式$\boldsymbol{\u03c7}$有界的？Aboulker、Duron、Jacob、Kimbrough、Thomassé及本文作者（2026）证明，弧集可表示为有限个可比有向图之并的锦标赛类满足该性质。那不具备此类分解的锦标赛类呢？Nguyen、Scott和Seymour（2025）提出的交叉锦标赛类便是这类例子，2026年的前述工作已证实这一点；本文通过改编Davies和McCarty（2021）以及Davies（2022）的方法，证明交叉锦标赛仍是多项式$\boldsymbol{\u03c7}$有界的。本文还进一步证明，无法将交叉锦标赛的这一结果推广到回边图为弦图的锦标赛。
英文摘要
Given a tournament $T$, Aboulker, Aubian, Charbit, and Lopes (2023) defined its clique number $\vecω(T)$ as the minimum clique number of a backedge graph of $T$, and raised the question: Which classes of tournaments are polynomially $\vecχ$-bounded? Aboulker, Duron, Jacob, Kimbrough, Thomassé, and this work's authors (2026) showed that this holds for classes of tournaments whose arc sets may be written as the union of a bounded number of comparability digraphs. What about classes of tournaments that do not admit such a decomposition? The crossing tournaments of Nguyen, Scott, and Seymour (2025) are an example of such a class, as shown in the aforementioned 2026 work; we show that nonetheless crossing tournaments are polynomially $\vecχ$-bounded by adapting a method of Davies and McCarty (2021) and Davies (2022). We additionally show that we cannot extend this result for crossing tournaments to tournaments with chordal graphs as backedge graphs.
