The uniform Hamilton-Waterloo Problem (HWP) asks for a resolvable (C M , C N )-decomposition of K v into α C M -factors and β C N -factors. We denote a solution to the uniform Hamilton Hamilton-Waterloo problem by HWP(v; M, N ; α, β). Our research concentrates on addressing some of the remaining unresolved cases, which pose a significant challenge to generalize. We place a particular emphasis on instances where the gcd(M, N ) = {2, 3}, with a specific focus on the parameter M = 6. We introduce modifications to some known structures, and develop new approaches to resolving these outstanding challenges in the construction of uniform 2-factorizations. This innovative method not only extends the scope of solved cases, but also contributes to a deeper understanding of the complexity involved in solving the Hamilton-Waterloo Problem.

The Hamilton-Waterloo problem asks for which s and r the complete graph K n can be decomposed into s copies of a given 2-factor F 1 and r copies of a given 2-factor F 2 (and one copy of a 1-factor if n is even). In this paper we generalize the problem to complete equipartite graphs K (n:m) and show that K (xyzw:m) can be decomposed into s copies of a 2-factor consisting of cycles of length xzm; and r copies of a 2-factor consisting of cycles of length yzm, whenever m is odd, s, r = 1, gcd(x, z) = gcd(y, z) = 1 and xyz = 0 (mod 4). We also give some more general constructions where the cycles in a given two factor may have different lengths. We use these constructions to find solutions to the Hamilton-Waterloo problem for complete graphs.

Abstract:We determine the distribution of 3-designs among the orbits of 4-and 5-element subsets under the action of PSLð2; 2 n Þ on the projective line. Thus we give complete information on all Kramer-Mesner matrices for the action of PSLð2; 2 n Þ on 3-sets versus 4-and 5-sets. As a consequence, all 3-designs with block sizes 4 and 5 and automorphism group PSLð2; 2 n Þ can immediately be obtained. #

Abstract:We determine the distribution of quadruple systems among the orbits of 4-element subsets under the action of PSL(2,q) on the projective line when q 1 (mod 4).

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