Cycles through an arc in regular 3-partite tournaments

摘要

A c-partite tournament is an orientation of a complete c-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament D is contained in an m-, (m + 1)- or (m + 2)-cycle for each m is an element of {3, 4,..., vertical bar V (D)vertical bar - 2}, and he also proved this conjecture for m = 3, 4, 5. In 2012, Xu et al. proved that every arc of a regular 3-partite tournament is contained in a 5- or 6-cycle, and in the same paper, the authors also posed the following conjecture: