Crossing tournaments are polynomially $\vecχ$-bounded

2026-08-10

AI Summary

This study investigates whether classes of tournaments that cannot be decomposed into a finite union of comparable oriented graphs—such as crossing tournaments—satisfy polynomial $\vec\chi$-boundedness. By refining and extending the structural graph-theoretic and coloring techniques of Davies and McCarty, and incorporating an analysis of backedge graphs, the authors establish for the first time that crossing tournaments are polynomially $\vec\chi$-bounded; that is, their chromatic number is bounded by a polynomial function of their clique number. Moreover, the work demonstrates that this property does not extend to the broader class of tournaments whose backedge graphs are chordal, thereby delineating the precise boundary of applicability for polynomial $\vec\chi$-boundedness in this context.

Abstract

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.

Problem

• tournaments
• polynomially χ-bounded
• crossing tournaments
• clique number
• backedge graph

Innovation

• crossing tournaments
• polynomially χ̃-bounded
• backedge graphs
• comparability digraphs
• chordal graphs