A minimum semi-degree sufficient condition for one-to-many disjoint path covers in semicomplete digraphs

Abstract

Let D be a digraph. We define the minimum semi-degree of D as δ0(D):=min⁡{δ+(D),δ−(D)}. Let k be a positive integer, and let S={s} and T={t1,…,tk} be any two disjoint subsets of V(D). A set of k internally disjoint paths joining source set S and sink set T that cover all vertices of D are called a one-to-many k-disjoint directed path cover (k-DDPC for short) of D. A digraph D is semicomplete if for every pair x,y of vertices of it, there is at least one arc between x and y.In this paper, we prove that every semicomplete digraph D of sufficiently large order n with δ0(D)≥⌈(n+k−1)/2⌉ has a one-to-many k-DDPC joining any disjoint source set S and sink set T, where S={s},T={t1,…,tk}.