The structure of $Δ(1, 2, 2)$-free tournaments

TL;DR

The paper delivers a complete structural description for $Δ(1,2,2)$-free tournaments by developing a backedge-graph framework and a constructive decomposition: every such tournament is either a basic tournament, a paving-tournament, or derived from a smaller $Δ(1,2,2)$-free by substituting a basic for a nice vertex or by a $P_7^-$-join on a bridge. Central to the approach are the basic tournaments $T_5$, $P_7^-$, and $P_7$, and the operations of substitution and join which preserve the $Δ(1,2,2)$-free property. The authors prove a backedge-graph theorem (components are monotone paths or $H_5$, $H_6$, $H_7$) and develop a triangle-reshuffling technique to realize paving orderings, enabling a full inductive construction. The results yield tight bounds for chromatic number, the size of the largest transitive subtournament, and the maximum number of vertex-disjoint cyclic triangles, advancing Erdős–Hajnal-type insights for tournaments and providing a practical, constructive framework for generating all $Δ(1,2,2)$-free tournaments.

Abstract

We extend the list of tournaments $S$ for which the complete structural description for tournaments excluding $S$ as a subtournament is known. Specifically, let $Δ(1, 2, 2)$ be a tournament on five vertices obtained from a cyclic triangle by substituting a two-vertex tournament for two of its vertices. In this paper, we show that tournaments excluding $Δ(1, 2, 2)$ as a subtournament are either isomorphic to one of three small tournaments, obtained from a transitive tournament by reversing edges in vertex-disjoint directed paths, or obtained from a smaller tournament with the same property by applying one of two operations. In particular, one of these operations creates a homogeneous set that induces a subtournament isomorphic to one of three fixed tournaments, and the other creates a homogeneous pair such that their union induces a subtournament isomorphic to a fixed tournament. As an application of this result, we present an upper bound for the chromatic number, a lower bound for the size of a largest transitive subtournament, and a lower bound for the number of vertex-disjoint cyclic triangles for such tournaments. The bounds that we present are all best possible.

Figures (12)

• Figure 1: Paw, diamond, claw, and bull.
• Figure 2: The tournament $\Delta(1, 2, 2)$ and its backedge graph with respect to the ordering $\tau = (u_1, \ldots, u_5)$.
• Figure 3: Three ordered graphs $H_5$, $H_6$, and $H_7$.
• Figure 4: Tournament $P_7^-$ and its backedge graph with respect to the ordering $\eta = (u_1, \ldots, u_6)$. $(\{u_1, u_2, u_3\}, \{u_4, u_5, u_6\})$ is the degree partition of $P_7^-$.
• Figure 5: Tournaments $T_7$, $U_7$, and $W_5$. Dashed edges indicated the ones reversed to obtain $U_7$ from $T_7$.

Theorems & Definitions (65)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5: Informal; see \ref{['Thm:triangle-packing']} for the formal statement
• Theorem 1.5: Informal; see \ref{['Thm:triangle-packing']} for the formal statement
• Theorem 1.6: Latka Latka03
• Theorem 1.7: Liu Liu12
• Theorem 1.8: Liu Liu15
• Theorem 2.1