A characterization of rich c-partite (c > 7) tournaments without (c + 2)-cycles

Abstract

Let c be an integer. A c-partite tournament is an orientation of a complete c-partite graph. A c-partite tournament is rich if it is strong, and each partite set has at least two vertices. In 1996, Guo and Volkmann characterized the structure of all rich c-partite tournaments without (c + 1)-cycles, which solved a problem by Bondy. They also put forward a problem that what the structure of rich c-partite tournaments without (c + k)-cycles for some k>1 is. In this paper, we answer the question of Guo and Volkmann for k = 2.

Figures (5)

• Figure 1: An example of the $8$-partite tournament with $|V_1|=3$ and $|V_j|=2$ for $2 \leq j \leq 8$. Here, the arcs between $v_1$ and other vertices are arbitrary.
• Figure 2: An example of $Q_m^1$. Here, all other possible arcs are of the same direction as the path.
• Figure 3: A cycle of length at most $(c+2)$ in $D[P\cup \{x\}]$ which contains two vertices in $V_1$ and two vertices in $V_2$.
• Figure 4: The structure of $D[P\cup \{x\}]$ of Type I -- Type IV in Claim \ref{['clm3']}.
• Figure 5: The structure of $D[P\cup \{x\}]$ of Proposition \ref{['property1']}(iv).

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Claim 2.1