REGULAR FACTORS IN POWERS OF CONNECTED GRAPHS

vertex set and the edge set of G, respectively. The number |V(G) | is called the order of G. If W £ V(G), then we denote by < JF>G the subgraph of G induced by W. If a spanning subgraph F of G is a regular graph of a degree m _ 0, then we say that F is an m-factor of G. For every integer n = 1, by the n-th power Gn of G we mean the graph with V(Gn) = V(G) and E(Gn) = {uv; u,ve V(G) with the property that 1 = dG(u, v) = n}, where dG denotes the distance between vertices in G. If n _ 1 is an odd integer and G has an n-factor, then the order of G is even. Chartrand, Polimeni and Stewart [2] and Sumner [6] proved that if G is a con-nected graph of an even order, then G2 has a 1-factor. Nebesky [4] proved that if G is a connected graph of an even order = 4, then G4 has a 3-factor. In the present paper these results will be generalized for every odd integer n ̂ 1. We shall prove the following theorem: Theorem 1. Let n ̂ 1 be an odd integer, and let G be a connected graph of an even order p = n + 1. Then Gn+1 has an n-factor. In the present paper we shall prove one more theorem, which complement