Characterizing Large Clique Number in Tournaments

TL;DR

The paper studies the tournament clique number $\overrightarrow{\omega}(T)$, defined as the minimum clique size among backedge graphs, and shows that large clique number is witnessed by a bounded-size subtournament from either the $A_n$ or $D_n$ family. By organizing the proof around an inductive bound on $\omega_A(T)+\omega_D(T)$ and introducing mountains, bag-chains, and zone decompositions, the authors prove a function $f$ with $\overrightarrow{\omega}(T) < f(\omega_A(T)+\omega_D(T))$ for all tournaments. This yields that for any fixed $n$, $\{A_n, D_n\}$-free tournaments have bounded clique number, and that both $A_n$ and $D_n$ subtournaments are necessary for unboundedness. The result advances understanding of unavoidable subtournaments in large-clique tournaments and connects to dichromatic number phenomena explored by Kim–Kim and Aboulker et al., by showing that large clique number can be certified by a bounded-size witness from these two families.

Abstract

Aboulker, Aubian, Charbit, and Lopes (2023) defined the clique number of a tournament to be the minimum clique number of one of its backedge graphs. Here we show that if $T$ is a tournament of sufficiently large clique number, then $T$ contains a subtournament of large clique number from one of two simple families of tournaments. In particular, large clique number is always certified by a bounded-size set. This answers a question of Aboulker, Aubian, Charbit, and Lopes (2023), and gives new insight into a line of research initiated by Kim and Kim (2018) into unavoidable subtournaments in tournaments with large dichromatic number.

Figures (11)

• Figure 1: The backedge graph of $A_n$ with respect to the ordering $<_B = (v_1, V(T_1), \dots, v_{n-1}, V(T_{n-1}), v_n)$. Each of $T_1, \dots, T_{n-1}$ is isomorphic to $A_{n-1}$.
• Figure 2: The tournament $D_n$
• Figure 3: A $(r,s)$-mountain where $K$ is a $(r,s)$-clique and $M_{uv}$ is the $r$-mountain which witnesses that $uv$ is $r$-heavy.
• Figure 4: Proof of Lemma \ref{['lem:2a']}.
• Figure 5: Proof of Lemma \ref{['lem:part2b1']}.

Theorems & Definitions (46)

• Lemma 1
• Proof 1: Proof
• Definition 1
• Definition 2
• Lemma 2
• Proof 2: Proof
• Theorem 3
• Corollary 4
• Theorem 5: Harutyunyan, Le, Thomassé, Wu harutyunyan2019coloring
• Corollary 6