On cycles through two arcs in strong multipartite tournaments

TL;DR

This paper proves that in strongly connected multipartite tournaments, there are at least two arcs that belong to directed cycles of every length from 3 up to the number of parts, extending previous results.

Contribution

It establishes the existence of two arcs with the cycle property in strongly connected multipartite tournaments, improving upon prior work that identified only one such arc.

Findings

Existence of two arcs with the cycle property in strongly connected multipartite tournaments.

Extension of previous results from one to two arcs.

Supports conjecture on multiple arcs with cycle properties.

Abstract

A multipartite tournament is an orientation of a complete $c$-partite graph. In [L. Volkmann, A remark on cycles through an arc in strongly connected multipartite tournaments, Appl. Math. Lett. 20 (2007) 1148--1150], Volkmann proved that a strongly connected $c$-partite tournament with $c \ge 3$ contains an arc that belongs to a directed cycle of length $m$ for every $m \in \{3, 4, \ldots, c\}$. He also conjectured the existence of three arcs with this property. In this note, we prove the existence of two such arcs.