Decompositions of complete uniform multi-hypergraphs into Berge paths and cycles of arbitrary lengths

Abstract

In 1981, Alspach conjectured that the complete graph $ K_{n} $ could be decomposed into cycles of arbitrary lengths, provided that the obvious necessary conditions would hold. This conjecture was proved completely by Bryant, Horsley and Pettersson in 2014. Moreover, in 1983, Tarsi conjectured that the obvious necessary conditions for packing pairwise edge-disjoint paths of arbitrary lengths in the complete multigraphs were also sufficient. The conjecture was confirmed by Bryant in 2010. In this paper, we investigate an analogous problem as the decomposition of the complete uniform multi-hypergraph $ \mu K_{n}^{(k)} $ into Berge cycles and Berge paths of arbitrary given lengths. We show that for every integer $ \mu\geq 1 $, $ n\geq 108 $ and $ 3\leq k<n $, $ \mu K_{n}^{(k)} $ can be decomposed into Berge cycles and Berge paths of arbitrary lengths, provided that the obvious necessary conditions hold, thereby generalizing a result by K\"{u}hn and Osthus on the decomposition of $K_{n}^{(k)}$ into Hamilton Berge cycles. Furthermore, we obtain the necessary and sufficient conditions for packing the cycles of arbitrary lengths in the complete multigraphs.