Cycles through an arc in regular 3-partite tournaments

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• 作者：Qiaoping Guo ; ^{guoqp@sxu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Linan Cui ; Wei Meng
• 关键词：Multipartite tournament ; Regular multipartite tournament ; 3-partite tournament ; Cycle
• 刊名：Discrete Mathematics
• 出版年：2016
• 出版时间：6 January 2016
• 年：2016
• 卷：339
• 期：1
• 页码：17-20
• 全文大小：355 K

文摘

A c$c$-partite tournament is an orientation of a complete c$c$-partite graph. In 2006, Volkmann conjectured that every arc of a regular 3-partite tournament D$D$ is contained in an m$m$-, (m+1)$(m+1)$- or (m+2)$(m+2)$-cycle for each m∈{3,4,…,|V(D)|−2}$m\in \{3,4,\dots ,\left|V\left(D\right)\right|-2\}$, and he also proved this conjecture for m=3,4,5$m=3,4,5$. 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$D$ is an e793e70654259" title="Click to view the MathML source">r$r$-regular 3-partite tournament with r≥2$r\ge 2$, then every arc of D$D$ is contained in a 3k$3k$- or (3k+1)$(3k+1)$-cycle for k=1,2,…,r−1$k=1,2,\dots ,r-1$.

It is known that Conjecture 1 is true for k=1$k=1$. In this paper, we prove Conjecture 1 for k=2$k=2$, which implies that Volkmann’s conjecture for e79499fd39" title="Click to view the MathML source">m=6$m=6$ is correct.