an:06164632
Zbl 1264.05088
Bahmanian, Amin; Rodger, Chris
Embedding factorizations for 3-uniform hypergraphs
EN
J. Graph Theory 73, No. 1-2, 216-224 (2013).
00317616
2013
j
05C65 05C60 05C70
amalgamations; detachments; complete 3-uniform hypergraphs; embedding; factorization; decomposition
Summary: In this article, two results are obtained on a hypergraph embedding problem. The proof technique is itself of interest, being the first time amalgamations have been used to address the embedding of hypergraphs. The first result finds necessary and sufficient conditions for the embedding a hyperedge-colored copy of the complete 3-uniform hypergraph of order \(m\), \(K^{3}_m\), into an \(r\)-factorization of \(K^{3}_n\), providing that \(n>2m+(-1+\sqrt{8m^{2}-16m-7}/2\). The second result finds necessary and sufficient conditions for an embedding when not only are the colors of the hyperedges of \(K^{3}_m\) given, but also the colors of all the ``pieces'' of hyperedges on these \(m\) vertices are prescribed (the ``pieces'' of hyperedges are eventually extended to hyperedges of size 3 in \(K^{3}_n\) by adding new vertices to the hyperedges of size 1 and 2 during the embedding process). Both these results make progress toward settling an old question of Cameron on completing partial 1-factorizations of hypergraphs.