Sectionable Tournaments: Their topology and Coloring

TL;DR

This paper explores the topological and combinatorial properties of sectionable tournaments, describing their acyclic complexes and establishing bounds on their chromatic number based on their structure.

Contribution

It introduces the class of sectionable tournaments, characterizes their acyclic complexes using discrete Morse theory, and derives bounds on their chromatic number related to their dimension.

Findings

Full description of the cell-structure of acyclic complexes of sectionable tournaments.

Establishment of an upper bound on the chromatic number based on the complex's dimension.

Demonstration of the role of the complex's dimension in tournament coloring.

Abstract

We provide a detailed study of topological and combinatorial properties of sectionable tournaments. This class forms an inductively constructed family of tournaments grounded over simply disconnected tournaments, those tournaments whose fundamental groups of acyclic complexes are non-trivial. When $T$ is a sectionable tournament, we fully describe the cell-structure of its acyclic complex $Acy(T)$ by using the adapted machinery of discrete Morse theory for acyclic complexes of tournaments. In the combinatorial side, we demonstrate that the dimension of the complex $Acy(T)$ has a role to play. We prove that if $T$ is a $(2r+1)$-sectionable tournament and $d$ is the dimension of $Acy(T)$, then the (acyclic) chromatic number of $T$ satisfies $\chi(T)\leq 2 \left( 2-1/(r+1) \right)^{\log(d+1)}-1$ where the logarithm has two as its base.