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 $c'$ such that every regular $r$-balanced $c$-partite tournament has a strong partition with $c\geq c'$. Figueroa, Montellano-Ballesteros and Olsen showed the existence of $ST(r)$ for all $r\geq 2$ and proved that $5\leq ST(2)\leq 7$. 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.

Figures (6)

• Figure 1: Four relationships between two partite sets
• Figure 2: Suitable $G_{2,4}$
• Figure 3: Four kinds of 4-vertex tournaments
• Figure 4: Arcs incident to $V_5$
• Figure 5: Arcs between $V_6$ and $V(T^{\tau}_2)$

Theorems & Definitions (37)

• Theorem 1.1: figueroa2023partition
• Theorem 1.2
• Definition 2.1: control①
• Definition 2.2: control②
• Definition 2.3: control③, control④
• Lemma 2.4: Folklore
• Lemma 2.5: figueroa2023partition
• Lemma 2.7
• proof
• Lemma 2.8