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

TL;DR

This paper extends classical graph decomposition results to hypergraphs, proving that complete uniform multi-hypergraphs can be decomposed into Berge paths and cycles of arbitrary lengths under certain conditions.

Contribution

It generalizes known results from graphs to hypergraphs, establishing conditions for decompositions into Berge paths and cycles of arbitrary lengths.

Findings

Decomposition of complete uniform multi-hypergraphs into Berge cycles and paths is possible under necessary conditions.

Provides necessary and sufficient conditions for packing cycles of arbitrary lengths in complete multigraphs.

Generalizes results from graph decompositions to hypergraph settings.

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.