Uniformly resolvable decompositions of Kv in 1-factors and 4-stars

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. A uniformly resolvable {X, Y }-decomposition of the complete graph Kv is an edge decomposition of Kv into exactly r X-factors and s Y -factors. In this article we determine necessary and sufficient conditions for when the complete graph Kv has a uniformly resolvable decompositions into 1-factors and K1,4-factors

α-Resolvable λ-fold G-designs

A λ-fold G-design is said to be α-resolvable if its blocks can be partitioned into classes such that every class contains each vertex exactly α times. In this paper we study the existence problem of an α-resolvable λ-fold G-design oforder v in the case when G is any connected subgraph of K_4 and prove that the necessary conditions for its existence are also sufficient

On uniformly resolvable (C4,K1,3)(C_4,K_{1,3})-designs

In this paper we consider the uniformly resolvable decompositions of the complete graph $K_v$ minus a 1-factor $(K_v − I)$ into subgraphs where each resolution class contains only blocks isomorphic to the same graph. We completely determine the spectrum for the case in which all the resolution classes consist of either 4-cycles or 3-stars

Hamilton-Waterloo problem with triangle and C9 factors

The Hamilton-Waterloo problem and its spouse-avoiding variant for uniform cycle sizes asks if Kv, where v is odd (or Kv - F, if v is even), can be decomposed into 2-factors in which each factor is made either entirely of m-cycles or entirely of n-cycles. This thesis examines the case in which r of the factors are made up of cycles of length 3 and s of the factors are made up of cycles of length 9, for any r and s. We also discuss a constructive solution to the general (m,n) case which fixes r and s