Construction, Extension and Paths of Near-Homogeneous Tournaments

TL;DR

This paper introduces a new method to construct near-homogeneous tournaments with various vertex counts and proves their path extendability, broadening the class of tournaments known to have this property.

Contribution

It generalizes the concept of homogeneous tournaments to near-homogeneous ones with both odd and even vertices, and demonstrates their path extendability.

Findings

New construction method for near-homogeneous tournaments with 4t+1 vertices.

Extension of near-homogeneity definition to even-vertex tournaments.

Proof that near-homogeneous tournaments are path extendable.

Abstract

A homogeneous tournament is a tournament with $4t+3$ vertices such that every arc is contained in exactly $t+1$ cycles of length $3$. Homogeneous tournaments are the first class of tournaments that are proved to be path extendable, which means that every nonhamiltonian path $P$ in such a tournament $T$ can be extended to a path $P'$ with the same initial and terminal vertex and $V(P')=V(P)\cup \{u\}$ for a certain vertex $u\in V(T)\backslash V(P)$. In order to find more path extendable tournaments we study the generalization of homogeneous tournaments called near-homogeneous tournaments, in which every arc is contained in $t$ or $t+1$ cycles of length $3$. Near-homogeneity has been defined in tournaments with $4t+1$ vertices. In this paper, we raise a new method to construct near-homogeneous tournaments with $4t+1$ vertices. We then show that the definition of near-homogeneous tournament can be extended to tournaments with an even number of vertices. Finally we verify path extendability of near-homogeneous tournaments, thus expand the class of path extendable tournaments.