Partitioning The Edge Set of a Hypergraph Into Almost Regular Cycles

Abstract

A cycle of length $t$ in a hypergraph is an alternating sequence $v_1,e_1,v_2\dots,v_t,e_t$ of distinct vertices $v_i$ and distinct edges $e_i$ so that $\{v_i,v_{i+1}\}\subseteq e_i$ (with $v_{t+1}:=v_1$). Let $\lambda K_n^h$ be the $\lambda$-fold $n$-vertex complete $h$-graph. Let $\mathcal G=(V,E)$ be a hypergraph all of whose edges are of size at least $h$, and $2\leq c_1\leq \dots\leq c_k\leq |V|$. In order to partition the edge set of $\mathcal G$ into cycles of specified lengths $c_1, \dots, c_k$, an obvious necessary condition is that $\sum_{i=1}^k c_i=|E|$. We show that this condition is sufficient in the following cases: (i) $h\geq \max\{c_k, \lceil n/2 \rceil+1\}$; (ii) $\mathcal G=\lambda K_n^h$, $h\geq \lceil n/2 \rceil+2$; (iii) $\mathcal G=K_n^h$, $c_1= \dots=c_k:=c$, $c|n(n-1), n\geq 85$. In (ii), we guarantee that each cycle is almost regular. In (iii), we also solve the case where a "small" subset $L$ of edges of $K_n^h$ is removed.