Construction, extension and paths of near-homogeneous tournaments

A homogeneous tournament is a tournament with 4 t + 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 ) ∪ { u } for a certain vertex u ∈ V ( T ) ∖ 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 4 t + 1 vertices. In this paper, we raise a new method to construct near-homogeneous tournaments with 4 t + 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.