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1.
    
A certain recursive construction for biembeddings of Latin squares has played a substantial role in generating large numbers of nonisomorphic triangular embeddings of complete graphs. In this article, we prove that, except for the groups and C 4 , each Latin square formed from the Cayley table of an Abelian group appears in a biembedding in which the second Latin square has a transversal. Such biembeddings may then be freely used as ingredients in the recursive construction. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 20:81‐88, 2012  相似文献   

2.
    
Ryser conjectured that the number of transversals of a latin square of order n is congruent to n modulo 2. Balasubramanian has shown that the number of transversals of a latin square of even order is even. A 1‐factor of a latin square of order n is a set of n cells no two from the same row or the same column. We prove that for any latin square of order n, the number of 1‐factors with exactly n ? 1 distinct symbols is even. Also we prove that if the complete graph K2n, n ≥ 8, is edge colored such that each color appears on at most edges, then there exists a multicolored perfect matching. © 2004 Wiley Periodicals, Inc.  相似文献   

3.
    
It is well known that mutually orthogonal latin squares, or MOLS, admit a (Kronecker) product construction. We show that, under mild conditions, “triple products” of MOLS can result in a gain of one square. In terms of transversal designs, the technique is to use a construction of Rolf Rees twice: once to obtain a coarse resolution of the blocks after one product, and next to reorganize classes and resolve the blocks of the second product. As consequences, we report a few improvements to the MOLS table and obtain a slight strengthening of the famous theorem of MacNeish.  相似文献   

4.
It is proved that every n×n Latin square has a partial transversal of length at least nO(log2n). The previous papers proving these results (including one by the second author) not only contained an error, but were sloppily written and quite difficult to understand. We have corrected the error and improved the clarity.  相似文献   

5.
    
We prove that, with the single exception of the 2‐group C, the Cayley table of each Abelian group appears in a face 2‐colorable triangular embedding of a complete regular tripartite graph in an orientable surface. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 71–83, 2010  相似文献   

6.
    
In a latin square of order n , a k ‐plex is a selection of kn entries in which each row, column, and symbol occurs k times. A 1 ‐plex is also called a transversal. A k ‐plex is indivisible if it contains no c ‐plex for 0 < c < k . We prove that, for all n ≥ 4 , there exists a latin square of order n that can be partitioned into an indivisible ? n / 2 ?‐plex and a disjoint indivisible ? n / 2 ?‐plex. For all n ≥ 3 , we prove that there exists a latin square of order n with two disjoint indivisible ? n / 2 ?‐plexes. We also give a short new proof that, for all odd n ≥ 5 , there exists a latin square of order n with at least one entry not in any transversal. Such latin squares have no orthogonal mate. Copyright © 2011 Wiley Periodicals, Inc. J Combin Designs 19:304‐312, 2011  相似文献   

7.
    
In 1974 Cruse gave necessary and sufficient conditions for an r × s partial latin square P on symbols σ12,…,σt, which may have some unfilled cells, to be completable to an n × n latin square on symbols σ12,…,σn, subject to the condition that the unfilled cells of P must be filled with symbols chosen from {σt + 1t + 2,…,σn}. These conditions consisted of r + s + t + 1 inequalities. Hall's condition applied to partial latin squares is a necessary condition for their completion, and is a generalization of, and in the spirit of Hall's Condition for a system of distinct representatives. Cropper asked whether Hall's Condition might also be sufficient for the completion of partial latin squares, but we give here a counterexample to Cropper's speculation. We also show that the r + s + t + 1 inequalities of Cruse's Theorem may be replaced by just four inequalities, two of which are Hall inequalities for P (i.e. two of the inequalities which constitute Hall's Condition for P), and the other two are Hall inequalities for the conjugates of P. © 2011 Wiley Periodicals, Inc. J Combin Designs 19:268‐279, 2011  相似文献   

8.
本文讨论型为2^nu^1的有对称正交侣的带洞自正交拉丁方(HSOLSSOM(2^nu^1))的谱。证明当n≤9时,HSOLSSOM(2^nu^1)存在的充分必要条件是u为偶数且n≥3u/2+1;当n≥263时,若u为偶数且n≥2(u-2),则HSOLSSOM(2^nu^1)存在。  相似文献   

9.
    
In this article, we verify Dade's projective invariant conjecture for the symplectic group Sp4(2 n ) and the special unitary group SU4(22n ) in the defining characteristic, that is, in characteristic 2. Furthermore, we show that the Isaacs–Malle–Navarro version of the McKay conjecture holds for Sp4(2 n ) and SU4(22n ) in the defining characteristic, that is, Sp4(2 n ) and SU4(22n ) are good for the prime 2 in the sense of Isaacs, Malle, and Navarro.  相似文献   

10.
    
《Discrete Mathematics》2022,345(6):112852
  相似文献   

11.
A latin bitrade is a pair of partial latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same sets of symbols. This survey paper summarizes the theory of latin bitrades, detailing their applications to critical sets, random latin squares and existence constructions for latin squares.   相似文献   

12.
    
Ahuva C. Shkop 《代数通讯》2013,41(10):3813-3823
In this article, I will prove that assuming Schanuel's conjecture, an exponential polynomial with algebraic coefficients can have only finitely many algebraic roots. Furthermore, this proof demonstrates that there are no unexpected algebraic roots of any such exponential polynomial. This implies a special case of Shapiro's conjecture: if p(x) and q(x) are two exponential polynomials with algebraic coefficients, each involving only one iteration of the exponential map, and they have common factors only of the form exp (g) for some exponential polynomial g, then p and q have only finitely many common zeros.  相似文献   

13.
    
《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   

14.
Summary It is proved that if the nonempty intersection of bounded closed convex sets AnB is contained in (A + F)U(B+F) and one of the following holds true: (i) the space X is less-than-three dimensional, (ii) AUB is convex, (iii) F is a one-point set, then AnBCA+F or AnBCB+F (Theorems 2 and 3). Moreover, under some hypotheses the characterization of A and B such that AnB is a summand of AUB is given (Theorem 3).  相似文献   

15.
The basic necessary condition for the existence of a TD(5, ; v)-TD(5, ; u), namely v 4u, is shown to be sufficient for any 1, except when (v, u) = (6, 1) and = 1, and possibly when (v, u) = (10, 1) or (52, 6) and = 1. For the case = 1, 86 new incomplete transversal designs are constructed. Several construction techniques are developed, and some new incomplete TDs with block size six and seven are also presented.  相似文献   

16.
    
A critical set is a partial latin square that has a unique completion to a latin square, and is minimal with respect to this property. Let scs(n) denote the smallest possible size of a critical set in a latin square of order n. We show that for all n, . Thus scs(n) is superlinear with respect to n. We also show that scs(n) ≥ 2n?32 and if n ≥ 25, . © 2007 Wiley Periodicals, Inc. J Combin Designs 15: 269–282, 2007  相似文献   

17.
    
Using the technique of amalgamation‐detachment, we show that the complete equipartite multigraph can be decomposed into cycles of lengths (plus a 1‐factor if the degree is odd) whenever there exists a decomposition of into cycles of lengths (plus a 1‐factor if the degree is odd). In addition, we give sufficient conditions for the existence of some other, related cycle decompositions of the complete equipartite multigraph .  相似文献   

18.
    
We study equitable partitions of Latin‐square graphs and give a complete classification of those whose quotient matrix does not have an eigenvalue ?3.  相似文献   

19.
    
The purpose of this article is to study a categorification of the n-th tensor power of the spin representation of U(𝔰𝔬(7, ?)) by using certain subcategories and projective functors of the Bernstein–Gelfand–Gelfand (BGG) category of the complex Lie algebra 𝔤𝔩 n .  相似文献   

20.
    
E. Park 《代数通讯》2013,41(7):2184-2192
In this article, we construct examples of n-folds X carrying an ample line bundle A ∈ Pic X such that property N p fails for K X  + (n + 1 + p)A. This shows that the condition of Mukai's conjecture is optimal for every n ≥ 1 and p ≥ 0.  相似文献   

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