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

TL;DR

This paper establishes a minimum semi-degree condition in large semicomplete digraphs that guarantees the existence of a set of disjoint paths covering all vertices, connecting a single source to multiple sinks.

Contribution

It proves a new sufficient semi-degree condition for one-to-many disjoint path covers in semicomplete digraphs, extending previous results in directed graph theory.

Findings

Every large semicomplete digraph with semi-degree at least eil((n+k-1)/2) has the desired path cover.

The condition guarantees paths connecting a single source to multiple sinks covering all vertices.

The result applies to digraphs with sufficiently large order n.

Abstract

Let $D$ be a digraph. We define the minimum semi-degree of $D$ as $\delta^{0}(D) := \min \{\delta^{+}(D), \delta^{-}(D)\}$. Let $k$ be a positive integer, and let $S = \{s\}$ and $T = \{t_{1}, \dots ,t_{k}\}$ 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 $D$ 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 $\delta^{0}(D) \geq \lceil (n+k-1)/2\rceil$ has a one-to-many $k$-DDPC joining any disjoint source set $S$ and sink set $T$, where $S = \{s\}, T = \{t_{1}, \dots, t_{k}\}$.