Perfect matching and Hamilton cycle decomposition of complete balanced (k+1)-partite k-uniform hypergraphs

Abstract

• We generalize the perfect matching (Hamilton cycle) decomposition on complete graph and complete uniform hypergraph into complete ( k + 1 )-parts k -uniform hypergraph K k + 1 , n ( k ) . • We prove our results using construction methods, which means that we can indeed find a perfect matching (Hamilton tight cycle) decomposition by computer in polynomial time. • Our construction methods will use some simple number theory knowledge. Let K k + 1 , n ( k ) denote the complete balanced ( k + 1 ) -partite k -uniform hypergraph, whose vertex set consists of k + 1 parts, each has n vertices and whose edge set contains all the k -element subsets with no two vertices from one part. In this paper, we prove that if k ∣ n and ( n k , k ) = 1 , then K k + 1 , n ( k ) has a perfect matching decomposition; if ( n , k ) = 1 , then K k + 1 , n ( k ) has a Hamilton tight cycle decomposition. In both cases, we use constructive methods which imply that we also give a polynomial algorithm to find a perfect matching decomposition or a Hamilton tight cycle decomposition.