A minimum semi-degree sufficient condition for one-to-many disjoint path covers in semicomplete digraphs (2208.09313v1)

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 $S = {s}, T = {t_{1}, \dots, t_{k}}$.