Given nonnegative integers v,m,n,α,β, the Hamilton–Waterloo problem asks for a factorization of the complete graph Kv into α Cm‐factors and β Cn‐factors. Without loss of generality, we may assume that n≥m. Clearly, v odd, n,m≥3, m∣v, n∣v and α+β=(v−1)/2 are necessary conditions. To date results have only been found for specific values of m and n. In this paper, we show that for any integers n≥m, these necessary conditions are sufficient when v is a multiple of mn and v>mn, except possibly when β=1 or 3. For the case where v=mn we show sufficiency when β>(n+5)/2 with some possible exceptions. We also show that when n≥m≥3 are odd integers, the lexicographic product of Cm with the empty graph of order n has a factorization into α Cm‐factors and β Cn‐factors for every 0≤α≤n, β=n−α, with some possible exceptions.

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Every 1‐rotational solution of a classic or twofold Oberwolfach problem (OP) of order n is generated by a suitable 2‐factor (starter) of Kn or 2Kn, respectively. It is shown that any starter of a twofold OP of order n gives rise to a starter of a classic OP of order 2n−1 (doubling construction). It is also shown that by suitably modifying the starter of a classic OP, one may obtain starters of some other OPs of the same order but having different parameters. The combination of these two constructions leads to lots of new infinite classes of solvable OPs. Still more classes can be obtained with the help of a third construction making use of the possible gracefulness of a graph whose connected components are cycles and at most one path. As one of the many applications, Hilton and Johnson's [J London Math Soc, 64 (2001) 513–522] bound s≥5r−1 about the solvability of OP(r,s) is improved to s≥⌊r/4⌋+10 in the case of r even. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 483‐503, 2012

Let Kv∗ denote the complete graph Kv if v is odd and Kv−I, the complete graph with the edges of a 1‐factor removed, if v is even. Given nonnegative integers v,M,N,α,β, the Hamilton–Waterloo problem asks for a 2‐factorization of Kv∗ into α CM‐factors and β CN‐factors, with a Cℓ‐factor of Kv∗ being a spanning 2‐regular subgraph whose components are ℓ‐cycles. Clearly, M,N≥3, M∣v, N∣v and α+β=⌊v−12⌋ are necessary conditions. In this paper, we extend a previous result by the same authors and show that for any odd v≠MNgcd(M,N) the above necessary conditions are sufficient, except possibly when α=1, or when β∈{1,3}. Note that in the case where v is odd, M and N must be odd. If M and N are odd but v is even, we also show sufficiency but with further possible exceptions. In addition, we provide results on 2‐factorizations of the complete equipartite graph and the lexicographic product of a cycle with the empty graph.

A Steiner triple system of order v (briefly STS(v)) is 1-rotational under G if it admits G as an automorphism group acting sharply transitively on all but one point. The spectrum of values of v for which there exists a 1-rotational STS(v) under a cyclic, an abelian, or a dicyclic group, has been established in Phelps and Rosa (Discrete Math 33:57-66, 1981), Buratti (J Combin Des 9:215-226, 2001) and Mishima (Discrete Math 308:2617-2619, 2008), respectively. Nevertheless, the spectrum of values of v for which there exists a 1-rotational STS(v) under an arbitrary group has not been completely determined yet. This paper is a considerable step forward to the solution of this problem. In fact, we leave as uncertain cases only those for which we have v = ( p 3 − p)n + 1 ≡ 1 (mod 96) with p a prime, n ≡ 0 (mod 4), and the odd part of ( p 3 − p)n that is square-free and without prime factors congruent to 1 (mod 6).