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A double-star S q 1 , q 2 is the graph consisting of the union of two stars, K 1 , q 1 and K 1 , q 2 , together with an edge joining their centers. The spectrum for S q 1 , q 2 -designs, i.e., the set of all the n ∈ N such that an S q 1 , q 2 -design of the order n exists, is well-known when q 1 = q 2 = 2 . In this article, S 2 , 2 -designs satisfying additional properties are investigated. We determine the spectrum for S 2 , 2 -designs that can be transformed into (K 4 − e) -designs by a double squash (bi-squash) passing through middle designs whose blocks are copies of a bull (the graph consisting of a triangle and two pendant edges). Here, the use of the difference method enables obtaining cyclic decompositions and determining the spectrum for cyclic S 2 , 2 -designs that can be purely bi-squashed into cyclic (K 4 − e) -designs (the middle bull designs are also cyclic). [ABSTRACT FROM AUTHOR]

Let H be a connected subgraph of a graph G. An H-factor of G is a spanning subgraph of G whose components are isomorphic to H. Given a set H of mutually non-isomorphic graphs, a uniform H -factorization of G is a partition of the edges of G into H-factors for some H ∈ H . In this article, we give a complete solution to the existence problem of uniform H -factorizations of K n − I (the graph obtained by removing a 1-factor from the complete graph K n ) for H = { C h , S (C h) } , where C h is a cycle of length an even integer h ≥ 4 and S (C h) is the graph consisting of the cycle C h with a pendant edge attached to each vertex. [ABSTRACT FROM AUTHOR]

Let H be a connected subgraph of a graph G. An H-factor of G is a spanning subgraph of G whose components are isomorphic to H. Given a set H of mutually non-isomorphic graphs, a uniform H -factorization of G is a partition of the edges of G into H-factors for some H ∈ H . In this article, we give a complete solution to the existence problem for uniform (C k , P k + 1) -factorizations of K n − I in the case when k is even. [ABSTRACT FROM AUTHOR]

We study the completion problem for simple k-Latin rectangles, which are a special case of the generalized latin rectangles for which embedding theorems are given by Andersen and Hilton (1980) in "Generalized Latin rectangles II: Embedding", Discrete Mathematics 31(3). Here an alternative proof of those theorems are given for k-Latin rectangles in the "simple" case. More precisely, generalizing two classic results on the completability of partial Latin squares, we prove the necessary and sufficient conditions for a completion of a simple m x n k-Latin rectangle to a simple k-Latin square of order n and we show that if m ≥ n/2, any simple partial k-Latin square P of order m embeds in a simple k-Latin square L of order n. [ABSTRACT FROM AUTHOR]

Given a collection of graphs H, a uniformly resolvable H -design of order v is a decomposition of the edges of Kv into isomorphic copies of graphs from H (also called blocks) in such a way that all blocks in a given parallel class are isomorphic to the same graph from H. We consider the case H = {K1,2,K1,3} and prove that the necessary conditions for the existence of such designs are also sufficient. [ABSTRACT FROM AUTHOR]

Let H be a finite fully simple semihypergroup. We prove that for all hyperproducts P of elements in H, the subsemihypergroup P̂ generated by P is hypercyclic and ∣P∣ ≤ ∣ P̂ ∣ ≤ 2. [ABSTRACT FROM AUTHOR]