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Partitioning the edge set of a hypergraph into almost regular cycles. (English) Zbl 1402.05173
Summary: A cycle of length $$t$$ in a hypergraph is an alternating sequence $$v_1,e_1,v_2,\ldots,v_t,e_t$$ 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 $$2\leq c_1\leq\cdots\leq c_k\leq|V|$$. In order to partition the edge set of $$\mathcal{G}$$ into cycles of specified lengths $$c_1,\ldots,c_k$$, an obvious necessary condition is that $$\sum_{i=1}^kc_i=|E|$$. We show that this condition is sufficient in the following cases.
(R1) $$h\geq\max\{c_k,\lceil n/2\rceil+1\}$$.
(R2) $$\mathcal{G}=\lambda K_n^h,\;h\geq\lceil n/2\rceil+2$$.
(R3) $$\mathcal{G}=K_n^h,\;c_1=\cdots=c_k:=c,c|n(n-1),\;n\geq85$$.
In (R2), we guarantee that each cycle is almost regular. In (R3), we also solve the case where a “small” subset $$L$$ of edges of $$K^h_n$$ is removed.
##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C65 Hypergraphs 05C38 Paths and cycles 05C45 Eulerian and Hamiltonian graphs
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