Construction, Extension and Paths of Near-Homogeneous Tournaments

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.