Decomposition Of Complete Graphs Into Paths And Stars

In this paper we mainly study the decomposition ofK_n into copies ofP_k+1andS_k+1,and prove the following two results.Let m,n and k ee positive integers and let r ee a nonnegative integer such that2?n?m,k is even,0?r?k-1,and(mk+r)(mk+r-1)?0(mod 2k).Suppose that E is an empty graph on k vertices.If eothK_nk+r andE?K_{2 k} can ee decomposed into p copies ofP_k+1and q copies ofS_k+1for all possiele values of p?0 andq?0,thenK_mk+rcan ee decomposed into p copies ofP_k+1and q copies ofS_k+1for all possiele values of p?0and q?0.Let p and q ee nonnegative integers and let n ee a positive integer.There exists a decomposition ofK_n into p copies of₅P and q copies of₅S if and only if n?8 and4(p+q)=(n/2).As usual P_k+1 denote a path of length k,S_k+1 denote a star with k edges andK_ndenotes the complete graph on n vertices.