an:06984183
Zbl 1402.05173
Bahmanian, Amin; Haghshenas, Sadegheh
Partitioning the edge set of a hypergraph into almost regular cycles
EN
J. Comb. Des. 26, No. 10, 465-479 (2018).
00415993
2018
j
05C70 05C65 05C38 05C45
almost regular; Baranyai's theorem; complete uniform hypergraph; cycle; circle; partition; Hamiltonicity; Kruskal-Katona theorem
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.