On Tree-Factor Covered Graphs

Abstract

Let $T(n)$ be the set of all trees with at least one and no more than $n$ edges. A $T(n)$-factor of a graph $G$ is defined to be a spanning subgraph of $G$ each component of which is isomorphic to one of $T(n)$. If every ${\text{K}}_{1.\text{k}}$ subgraph of $G$ is contained in a $T(n)$-factor of $G$, then $G$ is said to be $T(n)$-factor $k$-covered. In this paper, we give a criterion for a graph to be a $T(n)$-factor $k$-covered graph.