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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 introduce a family of hypergroups, called weakly complete, generalizing the construction of complete hypergroups. Starting from a given group G, our construction prescribes the β -classes of the hypergroups and allows some hyperproducts not to be complete parts, based on a suitably defined relation over G. The commutativity degree of weakly complete hypergroups can be related to that of the underlying group. Furthermore, in analogy to the degree of commutativity, we introduce the degree of completeness of finite hypergroups and analyze this degree for weakly complete hypergroups in terms of their β -classes. [ABSTRACT FROM AUTHOR]

Hypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of size not larger than 5, apart of isomorphisms. [ABSTRACT FROM AUTHOR]

In this paper, we show a new construction of hypergroups that, under appropriate conditions, are complete hypergroups or non-complete 1-hypergroups. Furthermore, we classify the 1-hypergroups of size 5 and 6 based on the partition induced by the fundamental relation β. Many of these hypergroups can be obtained using the aforesaid hypergroup construction. [ABSTRACT FROM AUTHOR]

If X is a connected graph, then an X-factor of a larger graph is a spanning subgraph in which all of its components are isomorphic to X. Given a set Γ of pairwise non-isomorphic graphs, a uniformly resolvable Γ-decomposition of a graph G is an edge decomposition of G into X-factors for some graph X ∈ Γ . In this article we completely solve the existence problem for decompositions of K v -I into C n -factors and K 1 , n -factors in the case when n is even. [ABSTRACT FROM AUTHOR]

In every hypergroup, the equivalence classes modulo the fundamental relation β are the union of hyperproducts of element pairs. Making use of this property, we introduce the notion of height of a β -class and we analyze properties of hypergroups where the height of a β -class coincides with its cardinality. As a consequence, we obtain a new characterization of 1-hypergroups. Moreover, we define a hierarchy of classes of hypergroups where at least one β -class has height 1 or cardinality 1, and we enumerate the elements in each class when the size of the hypergroups is n ≤ 4 , apart from isomorphisms. [ABSTRACT FROM AUTHOR]