Solution on strong partition of $2$-balanced regular multipartite tournaments

Abstract

We call a partition of a $c$-partite tournament into tournaments of order $c$ is strong if each tournament is strongly connected. The strong partition number denoted as $ST(r)$, represents the minimum integer $c^{\prime }$ such that every regular $r$-balanced $c$-partite tournament has a strong partition with $c\ge c^{\prime }$. Figueroa, Montellano-Ballesteros and Olsen showed the existence of $ST(r)$ for all $r\ge 2$ and proved that $5\le ST(2)\le 7$. In this note, we establish that $ST(2)=6$ and we also show the unique $2$-balanced $5$-partite tournament which has no strong partition.

10 pages, 6 figures