Partitioning The Edge Set of a Hypergraph Into Almost Regular Cycles

TL;DR

This paper establishes conditions under which the edges of certain hypergraphs can be partitioned into cycles of specified lengths, extending known results to more general hypergraph classes and ensuring almost regularity of cycles.

Contribution

It provides new sufficient conditions for hypergraph edge partitions into cycles of given lengths, including cases with almost regular cycles and hypergraphs with removed edges.

Findings

Partitioning is possible when the sum of cycle lengths equals the number of edges.

Conditions depend on minimum edge size and hypergraph structure.

Almost regular cycles can be guaranteed in certain hypergraph partitions.

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.