Construction, Extension and Paths of Near-Homogeneous Tournaments
Rongxia Tang, Zhaojun Chen, Zan-Bo Zhang
TL;DR
This work broadens the family of path extendable tournaments by introducing near-homogeneous tournaments, including a novel construction of order $8t+5$ from a homogeneous tournament and its complement. It extends the concept to even orders, provides structural results for orders $4t+2$ and rules out a straightforward extension to $4t$, and proves that all near-homogeneous tournaments of orders $4t+1$ and $4t+2$ are path extendable (with base cases verified computationally). The analysis blends 3-cycle containment counts (3-cyclic-index), almost-regularity classifications, and path-extension arguments, leveraging known results on arc-3-cyclic connectivity to establish path extendability across broad parameter ranges. Overall, the paper significantly expands the catalog of path extendable tournaments and offers constructive techniques and open questions for further exploration of near-homogeneous symmetry in tournaments.
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.