Cycles through an arc in regular 3-partite tournaments

A cc-partite tournament is an orientation of a complete cc-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament DD is contained in an mm-, (m+1)(m+1)- or (m+2)(m+2)-cycle for each m∈{3,4,…,|V(D)|−2}m∈{3,4,…,|V(D)|−2}, and he also proved this conjecture for m=3,4,5m=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:Conjecture 1. If DD is an rr-regular 3-partite tournament with r≥2r≥2, then every arc of DD is contained in a 3k3k- or (3k+1)(3k+1)-cycle for k=1,2,…,r−1k=1,2,…,r−1.It is known that Conjecture 1 is true for k=1k=1. In this paper, we prove Conjecture 1 for k=2k=2, which implies that Volkmann’s conjecture for m=6m=6 is correct.