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.

Generalized t-designs, which form a common generalization of objects such as t-designs, resolvable designs and orthogonal arrays, were defined by Cameron [P.J. Cameron, A generalisation of t-designs, Discrete Math. 309 (2009), 4835-4842]. In this paper, we define a related class of combinatorial designs which simultaneously generalize packing designs and packing arrays. We describe the sometimes surprising connections which these generalized designs have with various known classes of combinatorial designs, including Howell designs, partial Latin squares and several classes of triple systems, and also concepts such as resolvability and block colouring of ordinary designs and packings, and orthogonal resolutions and colourings. Moreover, we derive bounds on the size of a generalized packing design and construct optimal generalized packings in certain cases. In particular, we provide methods for constructing maximum generalized packings with t = 2 and block size k = 3 or 4.

We investigate the cop number of graphs based on combinatorial designs. Incidence graphs, point graphs, and block intersection graphs are studied, with an emphasis on finding families of graphs with large cop number. We generalize known results on Meyniel extremal families by supplying bounds on the incidence graphs of transversal designs, certain G-designs, and BIBDs with λ ≥ 1. Families of graphs with diameter 2, C 4 -free, and with unbounded chromatic number are described with the conjectured asymptotically maximum cop number.

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.

An ℓ-cycle system F of a graph Γ is a set of ℓ-cycles which partition the edge set of Γ. Two such cycle systems F and F ′ are said to be orthogonal if no two distinct cycles from F ∪ F ′ share more than one edge. Orthogonal cycle systems naturally arise from face 2-colourable polyehdra and in higher genus from Heffter arrays with certain orderings. A set of pairwise orthogonal ℓ-cycle systems of Γ is said to be a set of mutually orthogonal cycle systems of Γ.Let µ(ℓ, n) (respectively, µ ′ (ℓ, n)) be the maximum integer µ such that there exists a set of µ mutually orthogonal (cyclic) ℓ-cycle systems of the complete graph K n . We show that if ℓ ≥ 4 is even and n ≡ 1 (mod 2ℓ), then µ ′ (ℓ, n), and hence µ(ℓ, n), is bounded below by a constant multiple of n/ℓ 2 . In contrast, we obtain the following upper bounds:√ 2; and µ ′ (ℓ, n) ≤ n − 3 when n ≥ 4. We also obtain computational results for small values of n and ℓ.