On Min–Max Pair in Tournaments

Abstract

Let T be a tournament of order $n\ge 3$. A pair of distinct vertices x, y of T is called a min–max pair if one of x and y is of minimum out-degree, while the other is of maximum out-degree. Let xy be an arc such that x, y is a min–max pair. We call xy a min–max arc if x has minimum out-degree, and max–min arc otherwise. We prove that if yx is a min–max arc, then there exists a hamiltonian path from x to y; if xy is a max–min arc, then there exists a hamiltonian path from x to y with the exception of a few cases.