Disjoint path covers of star graphs

Abstract

Given a graph G, let S and T be two vertex-disjoint subsets of equal size k of G. A k-disjoint path cover of G corresponding to S and T is the union of k vertex-disjoint paths among S and T that spans G. If every vertex of S should be joined to a prescribed vertex in T, it is defined to be paired, otherwise it is unpaired. Let $S{T}_n$ be a star graph with bipartition $V_0$ and $V_1$. Let $S\subseteq V_0$ and $T\subseteq V_1$ be two vertex subsets of equal size k. It is shown in this paper that $S{T}_n$ admits an unpaired k-disjoint path cover between S and T, where $k\le n-2$ and $n\ge 4$. In view of the degree of $S{T}_n$, this result is optimal.