Decomposing tournaments into comparability graphs

2026-06-05

AI Summary

This work investigates the relationship between the dichromatic number and structural properties of directed graphs, with a focus on tournaments and their polynomial χ-boundedness. Introducing a new parameter—the partial order decomposition number (pod)—which measures the minimum number of partial orders whose union covers the arc set, the study establishes theoretical connections among pod, the dichromatic number, and the directed clique number. It proves that for any digraph \( D \), \( \text{dic}(D) \leq \omega(D)^{\text{pod}(D)} \), thereby demonstrating that digraph classes with bounded pod are polynomially χ-bounded. This result confirms a conjecture by Gutowski and Rams, further implying that tournaments with bounded pod have bounded domination number and enabling applications to substitution-closed classes and poset tournaments.

Abstract

In this note, we introduce the \emph{partial order decomposition number} of a digraph $D$, denoted $pod(D)$, defined as the minimum integer $k$ such that $A(D)=A(P_1)\cup\cdots\cup A(P_k)$, where $P_1,\ldots,P_k$ are partial orders on $V(D)$. We prove that $\dic(D)\le \diomega(D)^{pod(D)}$ for every digraph $D$. In particular, every class of digraphs with bounded $pod$ is polynomially $\dic$-bounded. We apply this to tournaments, showing that if $\mathcal C$ is a class of tournaments with bounded dichromatic number, then the closure of $\mathcal C$ under substitution is polynomially $\dic$-bounded, thereby making progress on a question of Aubian, Charbit, Lopes, and the first author. As further applications of $pod$, we prove that poset tournaments of bounded dimension are $\dic$-bounded, derive polynomial lower bounds on the directed clique number of an explicit family of tournaments, thereby answering a conjecture of Gutowski and Rams, and show that tournaments with bounded $pod$ have bounded domination number.

Problem

• partial order decomposition
• tournaments
• dichromatic number
• comparability graphs
• digraphs

Innovation

• partial order decomposition number
• dichromatic number
• tournaments
• polynomial dic-boundedness
• comparability graphs