Cycle Extendability in Extended Tournaments

Abstract

An extended tournament is an orientation of a $c (\ge 2)$-partite complete graph such that if there is one arc between two partite sets, say from $V_i$ to $V_j$, then there is an arc from every vertex in $V_i$ to every vertex in $V_j$. In this paper, we show the following. Let T be an extended tournament with $c\ge 4$ partite sets. For every non-Hamiltonian cycle C in T there is a cycle $C^\prime$ such that $V (C) \subset V (C^\prime )$ and $|C^\prime |\le |C| + 2$ if and only if T is Hamiltonian. Moreover, we give two examples to prove that the condition $c\ge 4$ is needed, and there exists an infinite family of extended tournaments which are Hamiltonian but have no cycle $C^\prime$ such that $V (C) \subset V (C^\prime )$ and $|C^\prime |= |C| + 1$ for a non-Hamiltonian cycle C in T.