Cycles through an arc 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 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:Conjecture 1. If D is an r-regular 3-partite tournament with r≥2, then every arc of D is contained in a 3k- or (3k+1)-cycle for k=1,2,…,r−1.It is known that Conjecture 1 is true for k=1. In this paper, we prove Conjecture 1 for k=2, which implies that Volkmann’s conjecture for m=6 is correct.