Enclosings of decompositions of complete multigraphs in 2-edge-connected r-factorizations

Abstract

A decomposition of a multigraph G is a partition of its edges into subgraphs G(1), . . . , G(k). It is called an r-factorization if every G(i) is r-regular and spanning.If G is a subgraph of H, a decomposition of G is said to be enclosed in a decomposition of H if, for every 1 ≤ i ≤ k, G(i) is a subgraph of H(i).Feghali and Johnson gave necessary and sufficient conditions for a given decomposition of λK n to be enclosed in some 2-edge-connected r-factorization of µK m for some range of values for the parameters n, m, λ, µ, r: r = 2, µ > λ and either m ≥ 2n − 1, or m = 2n − 2 and µ = 2 and λ = 1, or n = 3 and m = 4. We generalize their result to every r ≥ 2 and m ≥ 2n − 2. We also give some sufficient conditions for enclosing a given decomposition of λK n in some 2-edge-connected r-factorization of µK m for every r ≥ 3 and m = (2 − C)n, where C is a constant that depends only on r, λ and µ.

Embedding connected factorizations

“…Any partial 1-factorization of any sub-hypergraph of K 3 8 can be extended to an r-factorization of K 3 30 for each r P t2, 7, 14, 29, 58u. Any partial 1-factorization of any sub-hypergraph of K 4 8 can be extended to an r-factorization of K 4 45 for each r P t4, 28, 44, 172, 308u. Any partial 1-factorization of any sub-hypergraph of K 5 8 can be extended to an r-factorization of K 5 60 for each r P t2, 7, 14, 19, 29u.…”

“…‚ Last but not least, enclosing decompositions [4,19,30,31], and highly edge-connected factorizations of complete graphs [27,25] have been studied. In the absence of any corresponding results for hypergraphs, we pose the following problems.…”