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 $2\le c_1\le \cdots \le c_k\le |V|$. 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) $h\ge \max \{c_k,\lceil n/2\rceil +1\}$; (ii) $\mathcal{G}=\lambda K_n^h$, $h\ge \lceil n/2\rceil +2$; (iii) $\mathcal{G}=K_n^h$, $c_1=\cdots =c_k:=c$, $c|n(n-1),n\ge 85$. 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.

17 pages