A digraph $D$ with $n$ vertices is Hamiltonian (pancyclic and vertex‐pancyclic, respectively) if $D$ contains a Hamilton cycle (a cycle of every length $3,4,\text{…},n$, for every vertex $v\in V\left(D\right)$, a cycle of every length $3,4,\text{…},n$ through $v$, respectively.) It is well‐known that a strongly connected tournament is Hamiltonian, pancyclic, and vertex pancyclic. A digraph $D$ is cycle extendable if for every non‐Hamiltonian cycle $C$ of $D$, there is a cycle $C\prime$ such that $C\prime$ contains all vertices of $C$ plus another vertex of $D$. A cycle extendable digraph is fully cycle extendable if for every vertex $v\in V\left(D\right)$, there exists a cycle of length 3 through $v$. Note that full cycle extendability is a stronger property than vertex pancyclicity. While it is well‐known that every strongly connected tournament is vertex pancyclic, Hendry showed that not every strongly connected tournament is fully cycle extendable and characterized an infinite class of strongly connected tournaments, which are not fully cycle extendable. A $k$‐partite tournament is an orientation of a $k$‐partite complete graph (for $k=2$, it is called a bipartite tournament). Gutin and later Häggkvist and Manoussakis characterized Hamiltonian bipartite tournaments. A bipartite digraph $D$ with $n$ vertices is even pancyclic (even vertex pancyclic, respectively) if $D$ contains a cycle of every even length $4,6,\text{…},n$ (a cycle of every even length $4,6,\text{…},n$ through $v$ for every $v\in V\left(D\right)$, respectively). Beineke and Little, and Zhang proved that every bipartite tournament is even pancyclic and even vertex pancyclic, respectively, if and only if it is Hamiltonian and does not belong to a well‐defined infinite class of regular bipartite tournaments. We prove that unlike in the case of tournaments, every even pancyclic bipartite tournament is fully cycle extendable. We show that this result cannot be extended to $k$‐partite tournaments for any fixed $k\ge 3$ (where we naturally replace even vertex pancyclicity by vertex pancyclicity).

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多部分锦标赛中的汉密尔顿性，全周期性和全周期可扩展性

有向图 $d$ 与 $ñ$ 顶点是哈密顿量（分别是全圈和顶点全圈），如果 $d$ 包含一个汉密尔顿循环（每个长度的循环 $3，4，\text{…}，ñ$，对于每个顶点 $v\in V（d）$，每个长度的周期 $3，4，\text{…}，ñ$ 通过 $v$众所周知，一个紧密相连的锦标赛是哈密顿量，全循环量和顶点全循环量。有向图$d$ 如果每个非哈密顿循环都可扩展 $C$ 的 $d$，有一个周期 $C\prime$ 这样 $C\prime$ 包含的所有顶点 $C$ 再加上一个顶点 $d$。如果对于每个顶点，则循环可扩展有向图是完全循环可扩展的$v\in V（d）$，存在一个长度为3到 $v$。请注意，全周期可扩展性是比顶点泛环性更强的属性。众所周知，每个紧密联系的锦标赛都是顶点全循环的，但亨德利表明并非每个紧密联系的锦标赛都是完全循环可扩展的，并且具有无限类的紧密联系而不是完全循环可扩展的锦标赛。一种$ķ$业余锦标赛是一个方向 $ķ$局部完整图（用于 $ķ=2$，称为二分比赛）。Gutin以及后来的Häggkvist和Manoussakis代表了汉密尔顿双打锦标赛。二部图$d$ 与 $ñ$ 顶点即使是全循环的（分别是偶数全循环的顶点），如果 $d$ 包含每个偶数长度的循环 $4，6，\text{…}，ñ$ （每个偶数长度的周期 $4，6，\text{…}，ñ$ 通过 $v$ 每一个 $v\in V（d）$， 分别）。Beineke和Little以及Zhang证明，只要且仅当它是哈密顿量且不属于定义明确的无限定期常规锦标赛中，每个双向锦标赛都分别是全循环甚至顶点全循环。我们证明，与锦标赛不同，每场全周期两方锦标赛都是完全循环可扩展的。我们证明了这个结果不能扩展到$ķ$任何固定的部分锦标赛 $ķ\ge 3$ （在这里，我们自然用顶点全周期性替换了顶点全周期性）。