Decomposition of Complete Graphs into Paths and Stars with Different Number of Edges

Let P n and K n respectively denote a path and complete graph on n vertices. By a p H 1, q H 2 -decomposition of a graph G, we mean a decomposition of G into p copies of H 1 and q copies of H 2 for any admissible pair of nonnegative integers p and q, where H 1 and H 2 are subgraphs of G. In this paper, we show that for any admissible pair of nonnegative integers p and q, and positive integer n ≥ 4, there exists a p P 4, q S 4 -decomposition of K n if and only if 3 p + 4 q = (n 2), where S 4 is a star with 4 edges.