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1.
《组合设计杂志》2018,26(2):84-96
An array is row‐Latin if no symbol is repeated within any row. An array is Latin if it and its transpose are both row‐Latin. A transversal in an array is a selection of n different symbols from different rows and different columns. We prove that every Latin array containing at least distinct symbols has a transversal. Also, every row‐Latin array containing at least distinct symbols has a transversal. Finally, we show by computation that every Latin array of order 7 has a transversal, and we describe all smaller Latin arrays that have no transversal.  相似文献   

2.
Two Latin squares and , of even order n with entries , are said to be nearly orthogonal if the superimposition of L on M yields an array in which each ordered pair , and , occurs at least once and the ordered pair occurs exactly twice. In this paper, we present direct constructions for the existence of general families of three cyclic mutually orthogonal Latin squares of orders , , and . The techniques employed are based on the principle of Methods of Differences and so we also establish infinite classes of “quasi‐difference” sets for these orders.  相似文献   

3.
《组合设计杂志》2018,26(5):219-236
Let and . The integer partition of n is said to be realized if there is a latin square of order n with pairwise disjoint subsquares of order for each . In this paper, we construct latin squares realizing partitions of the form ; that is, partitions with s parts of size a and t parts of size b, where . Heinrich (1982) showed that (1) if and , then there is a latin square realizing , (2) is realized if and only if , and (3) is realized if and only if . In this paper, we resolve the open cases. We show that is realized if and only if and is realized if and only if .  相似文献   

4.
《组合设计杂志》2018,26(5):205-218
Let k, m, n, λ, and μ be positive integers. A decomposition of into edge‐disjoint subgraphs is said to be enclosed by a decomposition of into edge‐disjoint subgraphs if and, after a suitable labeling of the vertices in both graphs, is a subgraph of and is a subgraph of for all . In this paper, we continue the study of enclosings of given decompositions by decompositions that consist of spanning subgraphs. A decomposition of a graph is a 2‐factorization if each subgraph is 2‐regular and spanning, and is Hamiltonian if each subgraph is a Hamiltonian cycle. We give necessary and sufficient conditions for the existence of a 2‐factorization of that encloses a given decomposition of whenever and . We also give necessary and sufficient conditions for the existence of a Hamiltonian decomposition of that encloses a given decomposition of whenever and either or and , or , , and .  相似文献   

5.
《组合设计杂志》2018,26(6):267-279
In this paper, we derive the following bound on the size of a k‐wise L‐intersecting family (resp. cross L‐intersecting families) modulo a prime number:
  • (i) Let p be a prime, , and . Let and be two disjoint subsets of such that , or . Suppose that is a family of subsets of [n] such that for every and for every collection of k distinct subsets in . Then, This result may be considered as a modular version of Theorem 1.10 in [J. Q. Liu, S. G. Zhang, S. C. Li, H. H. Zhang, Eur. J. Combin. 58 (2016), 166‐180].
  • (ii) Let p be a prime, , and . Let and be two subsets of such that , or , or . Suppose that and are two families of subsets of [n] such that (1) for every pair ; (2) for every ; (3) for every . Then,
This result extends the well‐known Alon–Babai–Suzuki theorem to two cross L‐intersecting families.  相似文献   

6.
Let n and k be integers, with and . An semi‐Latin square S is an array, whose entries are k‐subsets of an ‐set, the set of symbols of S, such that each symbol of S is in exactly one entry in each row and exactly one entry in each column of S. Semi‐Latin squares form an interesting class of combinatorial objects which are useful in the design of comparative experiments. We say that an semi‐Latin square S is uniform if there is a constant μ such that any two entries of S, not in the same row or column, intersect in exactly μ symbols (in which case ). We prove that a uniform semi‐Latin square is Schur‐optimal in the class of semi‐Latin squares, and so is optimal (for use as an experimental design) with respect to a very wide range of statistical optimality criteria. We give a simple construction to make an semi‐Latin square S from a transitive permutation group G of degree n and order , and show how certain properties of S can be determined from permutation group properties of G. If G is 2‐transitive then S is uniform, and this provides us with Schur‐optimal semi‐Latin squares for many values of n and k for which optimal semi‐Latin squares were previously unknown for any optimality criterion. The existence of a uniform semi‐Latin square for all integers is shown to be equivalent to the existence of mutually orthogonal Latin squares (MOLS) of order n. Although there are not even two MOLS of order 6, we construct uniform, and hence Schur‐optimal, semi‐Latin squares for all integers . & 2012 Wiley Periodicals, Inc. J. Combin. Designs 00: 1–13, 2012  相似文献   

7.
Existence of LSs     
《组合设计杂志》2018,26(8):387-400
Let X be a ‐set and be a partition of X into n groups of size 2 and one group G0 of size 4. A large‐set‐plus denoted by LS is a partition of all ‐transverse triples of X into block sets of 3‐GDD(2n41)s with group set , and two block sets of 3‐GDD(2n)s with group set . In this paper, we study the existence problem of LSs and give a nearly complete solution.  相似文献   

8.
《组合设计杂志》2018,26(10):465-479
A cycle of length t in a hypergraph is an alternating sequence of distinct vertices and distinct edges so that (with ). Let be the λ‐fold n‐vertex complete h‐graph. Let be a hypergraph all of whose edges are of size at least h, and . In order to partition the edge set of into cycles of specified lengths , an obvious necessary condition is that . We show that this condition is sufficient in the following cases.
  • (R1) .
  • (R2) , .
  • (R3) , , .
In (R2), we guarantee that each cycle is almost regular. In (R3), we also solve the case where a “small” subset L of edges of is removed.  相似文献   

9.
《组合设计杂志》2018,26(10):480-486
In this paper, we show that if and , then there exists an almost resolvable k‐cycle system of order for all except possibly for and . Thus we give a partial solution to an open problem posed by Lindner, Meszka, and Rosa (J. Combin. Des., vol. 17, pp. 404–410, 2009).  相似文献   

10.
For a k‐subset X of , the set of differences on X is the set (mod n): . A conflict‐avoiding code CAC of length n and weight k is a collection of k‐subsets of such that = ? for any distinct . Let CAC() be the class of all the CACs of length n and weight k. The maximum size of codes in CAC(n, k) is denoted by . A code CAC(n, k) is said to be optimal if = . An optimal code is tight equi‐difference if = and each codeword in is of the form . In this paper, the necessary and sufficient conditions for the existence problem of optimal tight equi‐difference conflict‐avoiding codes of length n = and weight 3 are given. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 223–231, 2013  相似文献   

11.
《组合设计杂志》2018,26(4):154-173
Given a combinatorial design with block set , the block‐intersection graph (BIG) of is the graph that has as its vertex set, where two vertices and are adjacent if and only if . The i‐block‐intersection graph (i‐BIG) of is the graph that has as its vertex set, where two vertices and are adjacent if and only if . In this paper, several constructions are obtained that start with twofold triple systems (TTSs) with Hamiltonian 2‐BIGs and result in larger TTSs that also have Hamiltonian 2‐BIGs. These constructions collectively enable us to determine the complete spectrum of TTSs with Hamiltonian 2‐BIGs (equivalently TTSs with cyclic 2‐intersecting Gray codes) as well as the complete spectrum for TTSs with 2‐BIGs that have Hamilton paths (i.e. for TTSs with 2‐intersecting Gray codes). In order to prove these spectrum results, we sometimes require ingredient TTSs that have large partial parallel classes; we prove lower bounds on the sizes of partial parallel classes in arbitrary TTSs, and then construct larger TTSs with both cyclic 2‐intersecting Gray codes and parallel classes.  相似文献   

12.
《组合设计杂志》2018,26(2):51-83
Let denote the complete graph if v is odd and , the complete graph with the edges of a 1‐factor removed, if v is even. Given nonnegative integers , the Hamilton–Waterloo problem asks for a 2‐factorization of into α ‐factors and β ‐factors, with a ‐factor of being a spanning 2‐regular subgraph whose components are ℓ‐cycles. Clearly, , , and are necessary conditions. In this paper, we extend a previous result by the same authors and show that for any odd the above necessary conditions are sufficient, except possibly when , or when . 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.  相似文献   

13.
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 .  相似文献   

14.
《组合设计杂志》2018,26(1):12-26
A 1‐factorization of the complete multigraph is said to be indecomposable if it cannot be represented as the union of 1‐factorizations of and , where . It is said to be simple if no 1‐factor is repeated. For every and for every , we construct an indecomposable 1‐factorization of , which is not simple. These 1‐factorizations provide simple and indecomposable 1‐factorizations of for every and . We also give a generalization of a result by Colbourn et al., which provides a simple and indecomposable 1‐factorization of , where , , p prime.  相似文献   

15.
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.  相似文献   

16.
《组合设计杂志》2018,26(5):237-248
We establish that the logarithm of the number of latin d‐cubes of order n is and the logarithm of the number of sets of t ( is fixed) orthogonal latin squares of order n is . Similar estimations are obtained for systems of mutually strongly orthogonal latin d‐cubes. As a consequence, we construct a set of Steiner quadruple systems of order n such that the logarithm of its cardinality is as and .  相似文献   

17.
《组合设计杂志》2018,26(4):174-192
A cyclic code is a cyclic q‐ary code of length n, constant weight w and minimum distance d. Let denote the largest possible size of a cyclic code. The pure and mixed difference method plays an important role in the determination of upper bound on . By analyzing the distribution of odd mixed and pure differences, an improved upper bound on is obtained for . A new construction based on special sequences is provided and the exact value of is almost completely determined for all d and n except when and . Our constructions reveal intimate connections between cyclic constant weight codes and special sequences, particularly Skolem‐type sequences.  相似文献   

18.
《组合设计杂志》2018,26(11):547-559
Augmented orthogonal arrays (AOAs) were introduced by Stinson, who showed the equivalence between ideal ramp schemes and AOAs (Discrete Math. 341 (2018), 299–307). In this paper, we show that there is an AOA if and only if there is an OA which can be partitioned into subarrays, each being an OA, and that there is a linear AOA if and only if there is a linear maximum distance separable (MDS) code of length and dimension over , which contains a linear MDS subcode of length and dimension over . Some constructions for AOAs and some new infinite classes of AOAs are also given.  相似文献   

19.
《组合设计杂志》2018,26(8):369-386
Fisher proved in 1940 that any 2‐ design with has at least v blocks. In 1975, Ray‐Chaudhuri and Wilson generalized this result by showing that every t‐ design with has at least blocks. By combining methods used by Bose and Wilson in proofs of these results, we obtain new lower bounds on the size of t‐ coverings. Our results generalize lower bounds on the size of 2‐ coverings recently obtained by the first author.  相似文献   

20.
《组合设计杂志》2018,26(3):127-144
Steiner systems are a fascinating topic of combinatorics. The most studied Steiner systems are (Steiner triple systems), (Steiner quadruple systems), and . There are a few infinite families of Steiner systems in the literature. The objective of this paper is to present an infinite family of Steiner systems for all from cyclic codes. As a by‐product, many infinite families of 2‐designs are also reported in this paper.  相似文献   

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