On Tree-Factor Covered Graphs

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.