On the Hamilton–Waterloo Problem with Odd Orders

Given nonnegative integers , the Hamilton–Waterloo problem asks for a factorization of the complete graph into α ‐factors and β ‐factors. Without loss of generality, we may assume that . Clearly, v odd, , , and 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 , these necessary conditions are sufficient when v is a multiple of and , except possibly when or 3. For the case where we show sufficiency when with some possible exceptions. We also show that when are odd integers, the lexicographic product of with the empty graph of order n has a factorization into α ‐factors and β ‐factors for every , , with some possible exceptions.     

On the Existence of

Let there is an . For or , has been determined by Hanani, and for or , has been determined by the first author. In this paper, we investigate the case . A necessary condition for is . It is known that , and that there is an for all with a possible exception . We need to consider the case . It is proved that there is an for all with an exception and a possible exception , thereby, .     

Block‐Transitive and Point‐Primitive 2‐ Designs with Sporadic Socle

The purpose of this paper is to classify all pairs , where is a nontrivial 2‐ design, and acts transitively on the set of blocks of and primitively on the set of points of with sporadic socle. We prove that there exists only one such pair : is the unique 2‐(176,8,2) design and , the Higman–Sims simple group.     

On Flag‐Transitive Symmetric Designs of Affine Type

It is shown that, if is a nontrivial 2‐ symmetric design, with , admitting a flag‐transitive automorphism group G of affine type, then , p an odd prime, and G is a point‐primitive, block‐primitive subgroup of . Moreover, acts flag‐transitively, point‐primitively on , and is isomorphic to the development of a difference set whose parameters and structure are also provided.     

On the Distances between Latin Squares and the Smallest Defining Set Size

In this note, we show that for each Latin square L of order , there exists a Latin square of order n such that L and differ in at most cells. Equivalently, each Latin square of order n contains a Latin trade of size at most . We also show that the size of the smallest defining set in a Latin square is .     

Tight Heffter Arrays Exist for all Possible Values

A tight Heffter array is an matrix with nonzero entries from such that (i) the sum of the elements in each row and each column is 0, and (ii) no element from appears twice. We prove that exist if and only if both m and n are at least 3. If H has the property that all entries are integers of magnitude at most , every row and column sum is 0 over the integers, and H also satisfies ), we call H an integer Heffter array. We show integer Heffter arrays exist if and only if . Finally, an integer Heffter array is shiftable if each row and column contains the same number of positive and negative integers. We show that shiftable integer arrays exists exactly when both are even.     

Decomposition of Complete Graphs into Isomorphic Complete Bipartite Graphs

A decomposition of a complete graph into disjoint copies of a complete bipartite graph is called a ‐design of order n. The existence problem of ‐designs has been completely solved for the graphs for , for , K_{2, 3} and K_{3, 3}. In this paper, I prove that for all , if there exists a ‐design of order N, then there exists a ‐design of order n for all (mod ) and . Giving necessary direct constructions, I provide an almost complete solution for the existence problem for complete bipartite graphs with fewer than 18 edges, leaving five orders in total unsolved.     

The Hamilton–Waterloo Problem for ‐Factors and ‐Factors

The Hamilton–Waterloo problem asks for a 2‐factorization of (for v odd) or minus a 1‐factor (for v even) into ‐factors and ‐factors. We completely solve the Hamilton–Waterloo problem in the case of C₃‐factors and ‐factors for .     

Optimal Two‐Dimensional Optical Orthogonal Codes with the Best Cross‐Correlation Constraint

The study of optical orthogonal codes has been motivated by an application in an optical code‐division multiple access system. From a practical point of view, compared to one‐dimensional optical orthogonal codes, two‐dimensional optical orthogonal codes tend to require smaller code length. On the other hand, in some circumstances only with good cross‐correlation one can deal with both synchronization and user identification. These motivate the study of two‐dimensional optical orthogonal codes with better cross‐correlation than auto‐correlation. This paper focuses on optimal two‐dimensional optical orthogonal codes with the auto‐correlation and the best cross‐correlation 1. By examining the structures of w‐cyclic group divisible designs and semi‐cyclic incomplete holey group divisible designs, we present new combinatorial constructions for two‐dimensional ‐optical orthogonal codes. When and , the exact number of codewords of an optimal two‐dimensional ‐optical orthogonal code is determined for any positive integers n and .     

A Generalization of the Hamilton–Waterloo Problem on Complete Equipartite Graphs

The Hamilton–Waterloo problem asks for which s and r the complete graph can be decomposed into s copies of a given 2‐factor F₁ and r copies of a given 2‐factor F₂ (and one copy of a 1‐factor if n is even). In this paper, we generalize the problem to complete equipartite graphs and show that 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, , , and . 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.     

Optimal 1—Spontaneous Emission Error Designs*

A t‐spontaneous emission error design, denoted by t‐ SEED or t‐SEED in short, is a system of k‐subsets of a v‐set V with a partition of satisfying for any and , , where is a constant depending only on E. The design of t‐SEED was introduced by Beth et al. in 2003 (T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, M. Mussinger, Des Codes Cryptogr 29 (2003), 51–70) to construct quantum jump codes. The number m of designs in a t‐ SEED is called dimension, which corresponds to the number of orthogonal basis states in a quantum jump code. A t‐SEED is nondegenerate if every point appears in each of its member design. A nondegenerate t‐SEED is called optimal when it achieves the largest possible dimension. This paper investigates the dimension of optimal 1‐SEEDs, in which Baranyai's Lemma plays a significant role and the hypergraph distribution is closely related as well. Several classes of optimal 1‐SEEDs are shown to exist. In particular, we determine the exact dimensions of optimal 1‐ SEEDs for all orders v and block sizes k with .     

Skew Hadamard Difference Sets from Dickson Polynomials of Order 7

Skew Hadamard difference sets have been an interesting topic of study for over 70 years. For a long time, it had been conjectured the classical Paley difference sets (the set of nonzero quadratic residues in where ) were the only example in Abelian groups. In 2006, the first author and Yuan disproved this conjecture by showing that the image set of is a new skew Hadamard difference set in with m odd, where denotes the first kind of Dickson polynomials of order n and . The key observation in the proof is that is a planar function from to for m odd. Since then a few families of new skew Hadamard difference sets have been discovered. In this paper, we prove that for all , the set is a skew Hadamard difference set in , where m is odd and . The proof is more complicated and different than that of Ding‐Yuan skew Hadamard difference sets since is not planar in . Furthermore, we show that such skew Hadamard difference sets are inequivalent to all existing ones for by comparing the triple intersection numbers.     

Enumerating the Walecki‐Type Hamiltonian Cycle Systems

Let be the complete graph on v vertices. A Hamiltonian cycle system of odd order v (briefly ) is a set of Hamiltonian cycles of whose edges partition the edge set of . By means of a slight modification of the famous of Walecki, we obtain 2ⁿ pairwise distinct and we enumerate them up to isomorphism proving that this is equivalent to count the number of binary bracelets of length n, i.e. the orbits of , the dihedral group of order 2n, acting on binary n‐tuples.     

The Existence and Construction of (K5∖e)‐Designs of Orders 27, 135, 162, and 216

The problem of the existence of a decomposition of the complete graph into disjoint copies of has been solved for all admissible orders n, except for 27, 36, 54, 64, 72, 81, 90, 135, 144, 162, 216, and 234. In this paper, I eliminate 4 of these 12 unresolved orders. Let Γ be a ‐design. I show that divides 2^k3 for some and that . I construct ‐designs by prescribing as an automorphism group, and show that up to isomorphism there are exactly 24 ‐designs with as an automorphism group. Moreover, I show that the full automorphism group of each of these designs is indeed . Finally, the existence of ‐designs of orders 135, 162, and 216 follows immediately by the recursive constructions given by G. Ge and A. C. H. Ling, SIAM J Discrete Math 21(4) (2007), 851–864.     

Directed PBDs with Block Sizes from K Where K

The research on directed PBDs is motivated by the construction of t‐deletion/insertion‐correcting codes. Fuji‐Hara, Miao, Wang, and Yin have determined the existence of directed PBDs with block sizes from the set and the set completely. In this paper, we consider the cases of . We settle almost completely for these cases, leaving finite values undetermined.     

Latin Squares with a Unique Intercalate

Suppose that and . We construct a Latin square of order n with the following properties:

• has no proper subsquares of order 3 or more .
• has exactly one intercalate (subsquare of order 2) .
• When the intercalate is replaced by the other possible subsquare on the same symbols, the resulting Latin square is in the same species as .

Hence generalizes the square that Sade famously found to complete Norton's enumeration of Latin squares of order 7. In particular, is what is known as a self‐switching Latin square and possesses a near‐autoparatopism.     

A Complete Solution to the Existence of ‐Cycle Frames of Type

The existence problem of a ‐cycle frame of type is now solved for any quadruple .     

Flag‐Transitive 2‐ Symmetric Designs with Sporadic Socle

Let be a nontrivial 2‐ symmetric design admitting a flag‐transitive, point‐primitive automorphism group G of almost simple type with sporadic socle. We prove that there are up to isomorphism six designs, and must be one of the following: a 2‐(144, 66, 30) design with or , a 2‐(176, 50, 14) design with , a 2‐(176, 126, 90) design with or , or a 2‐(14,080, 12,636, 11,340) design with .     

Semiframes with Block Size Three and Odd Group Size

A ‐semiframe of type is a ‐GDD of type , , in which the collection of blocks can be written as a disjoint union where is partitioned into parallel classes of and is partitioned into holey parallel classes, each holey parallel class being a partition of for some . A ‐SF is a ‐semiframe of type in which there are p parallel classes in and d holey parallel classes with respect to . In this paper, we shall show that there exists a (3, 1)‐SF for any if and only if , , , and .     

Set Systems with L‐Intersections and k‐Wise L‐Intersecting Families

In this paper, by employing linear algebra methods we obtain the following main results:

• (i) Let and be two disjoint subsets of such that Suppose that is a family of subsets of such that for every pair and for every i. Then Furthermore, we extend this theorem to k‐wise L‐intersecting and obtain the corresponding result on two cross L‐intersecting families. These results show that Snevily's conjectures proposed by Snevily (2003) are true under some restricted conditions. This result also gets an improvement of a theorem of Liu and Hwang (2013).
• (ii) Let p be a prime and let and be two subsets of such that or and Suppose that is a family of subsets of [n] such that (1) for every pair (2) for every i. Then This result improves the existing upper bound substantially.