Some sufficient conditions for the existence of a 1‐factor

The following theorem is proved: Let G be a graph of even order. Assume that there exists a connected spanning subgraph F of G such that for every set U of four vertices in G , if the subgraph of F induced by U is a star, then the subgraph of G induced by U is complete. Then G has a 1‐factor. The above theorem is derived from another sufficient condition for the existence of a 1‐factor, which is also proved in this paper (Lemma 1).