On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity

The Hamilton-Waterloo problem asks for a decomposition of the complete graph of order v into r copies of a 2-factor F1 and s copies of a 2-factor F2 such that r+s = v−1 2 . If F1 consists of m-cycles and F2 consists of n cycles, we say that a solution to (m, n)- HWP(v; r, s) exists. The goal is to find a decomposition for every possible pair (r, s). In this paper, we show that for odd x and y, there is a solution to (2kx, y)-HWP(vm; r, s) if gcd(x, y) ≥ 3, m ≥ 3, and both x and y divide v, except possibly when 1 ∈ {r, s}

Orientable ℤ \u3c inf\u3e n -distance magic labeling of the Cartesian product of many cycles

The following generalization of distance magic graphs was introduced in [2]. A directed ℤn- distance magic labeling of an oriented graph G = (V,A) of order n is a bijection ℓ: V → ℤn with the property that there is a μ ∈ ℤn (called the magic constant) such that If for a graph G there exists an orientation G such that there is a directed ℤn-distance magic labeling ℓ for G, we say that G is orientable ℤn-distance magic and the directed ℤn-distance magic labeling ℓ we call an orientable ℤn-distance magic labeling. In this paper, we find orientable ℤn- distance magic labelings of the Cartesian product of cycles. In addition, we show that even-ordered hypercubes are orientable ℤn-distance magic

Fixed block configuration group divisible designs with block size six

AbstractWe present constructions and results about GDDs with two groups and block size six. We study those GDDs in which each block has configuration (s,t), that is in which each block has exactly s points from one of the two groups and t points from the other. We show the necessary conditions are sufficient for the existence of GDD(n,2,6;λ1,λ2)s with fixed block configuration (3,3). For configuration (1,5), we give minimal or near-minimal index examples for all group sizes n≥5 except n=10,15,160, or 190. For configuration (2,4), we provide constructions for several families of GDD(n,2,6;λ1,λ2)s

The 3-GDDs of type g3u2g^3u^2

A 3-GDD of type ${g^3u^2}$ exists if and only if $g$ and $u$ have the same parity, $3$ divides $u$ and $u\leq 3g$.Such a 3-GDD of type ${g^3u^2}$ is equivalent to an edge decomposition of $K_{g,g,g,u,u}$ into triangles

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

TS(v,λ) with Cyclic 2-Intersecting Gray Codes: v≡0 or 4(mod12)

A TS(v,λ) is a pair (V,B) where V contains v points and B contains 3-element subsets of V so that each pair in V appears in exactly λ blocks. A 2-block intersection graph (2-BIG) of a TS(v,λ) is a graph where each vertex is represented by a block from the TS(v,λ) and each pair of blocks Bi,Bj∈B are joined by an edge if |Bi∩Bj|=2. We show that there exists a TS(v,λ) for v≡0 or 4(mod12) whose 2-BIG is Hamiltonian for all admissible (v,λ). This is equivalent to the existence of a TS(v,λ) with a cyclic 2-intersecting Gray code

Mutually orthogonal equitable Latin rectangles

Let ab=n2. We define an equitable Latin rectangle as an a×b matrix on a set of n symbols where each symbol appears either bn⌉ or ⌊bn⌋ times in each row of the matrix and either an⌉ or ⌊an⌋ times in each column of the matrix. Two equitable Latin rectangles are orthogonal in the usual way. Denote a set of ka×b mutually orthogonal equitable Latin rectangles as a k MOELR (a,b;n). When a≠9,18,36, or 100, then we show that the maximum number of k MOELR (a,b;n) \u3c 3 for all possible values of (a,b). © 2011 Elsevier B.V. All rights reserved

The Hamilton-Waterloo Problem with 4-Cycles and a Single Factor of n-Cycles

A 2-factor in a graph G is a 2-regular spanning subgraph of G, and a 2-factorization of G is a decomposition of all the edges of G into edge-disjoint 2-factors. A {Cmr, Cns}-factorization of Kυasks for a 2-factorization of Kυ, where r of the 2-factors consists of m-cycles, and s of the 2-factors consists of n-cycles. This is a case of the Hamilton-Waterloo problem with uniform cycle sizes m and n. If υ is even, then it is a decomposition of Kυ- F where a 1-factor F is removed from Kυ. We present necessary and sufficient conditions for the existence of a {C4r, Cn1}-factorization of Kυ- F. © 2012 Springer Japan