Embedding Irregular Colorings into Connected Factorizations
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For r:=(r_1,…,r_k), an r-factorization of the complete λ-fold h-uniform n-vertex hypergraph λ K_n^h is a partition of (the edges of) λ K_n^h into F_1,…, F_k such that for i=1,…,k, F_i is r_i-regular and spanning. Suppose that n ≥ (h-1)(2m-1). Given a partial r-factorization of λ K_m^h, that is, a coloring (i.e. partition) P of the edges of λ K_m^h into F_1,…, F_k such that for i=1,…,k, F_i is spanning and the degree of each vertex in F_i is at most r_i, we find necessary and sufficient conditions that ensure P can be extended to a connected r-factorization of λ K_n^h (i.e. an r-factorization in which each factor is connected). Moreover, we prove a general result that implies the following. Given a partial s-factorization P of any sub-hypergraph of λ K_m^h, where s:=(s_1,…,s_q) and q is not too big, we find necessary and sufficient conditions under which P can be embedded into a connected r-factorization of λ K_n^h. These results can be seen as unified generalizations of various classical combinatorial results such as Cruse's theorem on embedding partial symmetric latin squares, Baranyai's theorem on factorization of hypergraphs, Hilton's theorem on extending path decompositions into Hamiltonian decompositions, Häggkvist and Hellgren's theorem on extending 1-factorizations, and Hilton, Johnson, Rodger, and Wantland's theorem on embedding connected factorizations.
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