On cycles in regular 3-partite tournaments

Abstract

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∈{3,4,…,|V(D)|−2}, and this conjecture was proved to be correct for 3≤m≤7. In 2012, Xu et al. conjectured that every arc of an r-regular 3-partite tournament D with r≥2 is contained in a (3k−1)- or 3k-cycle for k=2,3,…,r. They proved that this conjecture is true for k=2. In this paper, we confirm this conjecture for k=3, which also implies that Volkmann’s conjecture is correct for m=7,8.