多部竞赛图中包含某条弧的圈

On Cycles through an Arc in Multipartite Tournaments

多部竞赛图或n部竞赛图是指一个完全n部无向图的定向图.2007年Volkmann证明了每个强连通的n部竞赛图(n≥3)至少存在一条弧它包含在从3到n的每个长度的圈中.在此基础上给出了强连通n部竞赛图中存在一条弧它包含在从3到n+1的每个长度的圈中的一个充分条件,并举例说明该条件在某种意义上的最佳可能性.

A multipartite or n-partite tournament is an orientation of a complete n-partite graph. In 2007, Volkmann proved that every strong n-partite tournament with n ≥ 3 contains at least one arc that belongs to an m-cycle for each m C {3, 4, ... , n}. In this paper we give a sufficient condition for strong n-partite tournaments such that there exists at least one arc which belongs to an m-cycle for every rn C {3, 4,... , n＋ 1}. By some examples we illustrate that this condition is in some sense best possible.