Orthogonally Resolvable Cycle Decompositions

J of Combinatorial Designs

Danziger

Traetta

2017

Self Cite

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.

“…We use

[a, b]

to denote the set of integers from a to b inclusive; clearly,

[a, b]

is empty when

a > b

. We use the following notation from . Definition Given a positive integer N and a graph G ,

G [N]

is the lexicographic product of G with the empty graph on N vertices.…”

Section: Preliminariesmentioning

confidence: 99%

“…We use [ , ] to denote the set of integers from to inclusive; clearly, [ , ] is empty when > . We use the following notation from [18,19,36]. We will also use the concept of tensor products, which we now define.…”

Section: Graph Productsmentioning

confidence: 99%

On the Hamilton–Waterloo problem with odd cycle lengths

J of Combinatorial Designs

Danziger

Traetta

2017

Self Cite

“…1 respectively (these exist by Lemma 3.4 with h = m = 7, x = 2 and w 1 = w = 1 or 2). Since there exist GHD (8,18), GHD(9, 21), GHD (10,24), GHD (11,27) and GHD (13,33) each containing a sub-GHD(2, 0) (by Lemmas 4.9 and 4.10) the result follows.…”

Section: Existence Of Ghd(n + 2 3n)mentioning

confidence: 69%

“…There exists an H(s, 2n) if and only if n = 0 or n ≤ s ≤ 2n − 1, (s, 2n) ∈ {(2, 4), (3,4), (5,6), (5,8) is equivalent to a doubly resolvable balanced incomplete block design BIBD(v, k, λ). Doubly resolvable designs and related objects have been studied, for example, in [13,22,27,28,29,34,47,48]. In particular, Fuji-Hara and Vanstone investigated orthogonal resolutions in affine geometries, showing the existence of a doubly resolvable BIBD(q n , q, 1) for prime powers q and integers n ≥ 3.…”

Section: Theorem 12 (Mullin and Wallismentioning

confidence: 99%

See 1 more Smart Citation

On generalized Howell designs with block size three

Abel

Bailey

et al. 2015

Des. Codes Cryptogr.

In this paper, we examine a class of doubly resolvable combinatorial objects. Let t, k, λ, s and v be nonnegative integers, and let X be a set of v symbols. A generalized Howell design, denoted t-GHD k (s, v; λ), is an s × s array, each cell of which is either empty or contains a k-set of symbols from X, called a block, such that: (i) each symbol appears exactly once in each row and in each column (i.e. each row and column is a resolution of X); (ii) no t-subset of elements from X appears in more than λ cells. Particular instances of the parameters correspond to Howell designs, doubly resolvable balanced incomplete block designs (including Kirkman squares), doubly resolvable nearly Kirkman triple systems, and simple orthogonal multi-arrays (which themselves generalize mutually orthogonal Latin squares). Generalized Howell designs also have connections with permutation arrays and multiply constant-weight codes.In this paper, we concentrate on the case that t = 2, k = 3 and λ = 1, and write GHD(s, v). In this case, the number of empty cells in each row and column falls between 0 and (s−1)/3. Previous work has considered the existence of GHDs on either end of the spectrum, with at most 1 or at least (s − 2)/3 empty cells in each row or column. In the case of one empty cell, we correct some results of Wang and Du, and show that there exists a GHD(n + 1, 3n) if and only if n ≥ 6, except possibly for n = 6. In the case of two empty cells, we show that there exists a GHD(n + 2, 3n) if and only if n ≥ 6. Noting that the proportion of cells in a given row or column of a GHD(s, v) which are empty falls in the interval [0, 1/3), we prove that for any π ∈ [0, 5/18], there is a GHD(s, v) whose proportion of empty cells in a row or column is arbitrarily close to π.

On the Hamilton–Waterloo Problem with Odd Orders

J of Combinatorial Designs

Danziger

Traetta

2016

Self Cite

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.