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symbols there are four ordered pairs ði; jÞ such that a ij ¼ s h ; a 0 ij ¼ s k : We explore ways of constructing Hamiltonian double latin squares (HLS), symmetric HLS, sets of mutually orthogonal HLS ... A double latin square of order 2n on symbols s 1 ; y; s n is a 2n 2n matrix A ¼ ða ij Þ in which each a ij is one of the symbols s 1 ; y; s n and each s k occurs twice in each row and twice in each column ... In this paper we study double latin squares in which the occurrences of each symbol describe a Hamiltonian cycle. Such double latin squares are called Hamiltonian double latin squares. ...doi:10.1016/s0095-8956(02)00029-1 fatcat:geeyzeqos5gg3jj4iwrkvioy6y
The authors give constructions of Hamiltonian double Latin squares, symmetric Hamiltonian double Latin squares, and cyclic 05 COMBINATORICS 5164 symmetric Hamiltonian double Latin squares. ... A Hamiltonian double Latin square, denoted HLS(27), is a double Latin square in which each o-cycle is a Hamiltonian cycle. ...
In this paper the author obtains a necessary con- dition and a sufficient condition for constructing double magic squares by using orthogonal Latin squares, and constructs the smallest double magic squares ... The so-called double magic squares are “addition-multiplication magic squares”. ...
Lecture Notes in Computer Science
For construction of these paths, employing the Hamiltonian Circuit Latin Square(HCLS) we present O(n 2 ) parallel routing algorithm on circulant networks. ... Double-loop and 2-circulant networks are widely used in the design and implementation of local area networks and parallel processing architectures. ... Design of the Hamiltonian Circuit Latin Square to the Parallel Routing Algorithm on Circulant Networks Let A and B be any two nodes on G(n; ±s 1 ,±s 2 ,...,±s k ). ...doi:10.1007/11577188_22 fatcat:yxnzjhsmt5hrpdpfy36erxvuca
For construction of these paths, employing the Hamiltonian Circuit Latin Square(HCLS) we present O(n 2 ) parallel routing algorithm on circulant networks. ... Double-loop and 2-circulant networks are widely used in the design and implementation of local area networks and parallel processing architectures. ... Design of the Hamiltonian Circuit Latin Square to the Parallel Routing Algorithm on Circulant Networks Let A and B be any two nodes on G(n; ±s 1 ,±s 2 ,...,±s k ). ...doi:10.1016/j.ins.2005.12.014 fatcat:qfby6ahb45ekhhofxwgqalof5m
A Latin square is pan-Hamiltonian if every pair of rows forms a single cycle. Such squares are related to perfect 1-factorisations of the complete bipartite graph. ... A square is atomic if every conjugate is pan-Hamiltonian. These squares are indivisible in a strong sense – they have no proper subrectangles. ... A Latin square is pan-Hamiltonian if and only if it contains no proper subrectangles. In particular every pan-Hamiltonian square is N ∞ . ...doi:10.37236/1441 fatcat:vqj3dgmx5fal3g55e2s6s5zfiq
Fig. 2 is such a latin square of order 7. ... For the convenience in the proof of our main result, we shall use a special latin square M = [m i,j ] of order odd n which is a circulant latin square with 1st row (1, n+3 2 , 2, n+5 2 , 3, . . . , n+n ...doi:10.1016/j.disc.2008.07.018 fatcat:7xwsp7f22bfybfzikwubregqmq
More concretely, we show that the number of vertices is at least (L_n)^3/2-o(1), where L_n is the number of order-n Latin squares. ... It is well-known that an order-n Latin square may be viewed as a tristochastic array where every line contains n-1 zeros and a single 1 entry. ... The idea is to use a Hamiltonian double Latin square X to define the top n 2 layers of A. ...arXiv:1208.4218v1 fatcat:2blffg4eprfxpaoqnw5l6de5ua
A product construction is presented for building pairs of orthogonal latin squares such that one member of the pair has a certain hamiltonian property. ... In part I, we explore a connection between orthogonal latin squares and embeddings. ... Acknowledgements The first author would like to thank Wendy Myrvold for helpful discussions regarding latin squares. ...doi:10.1002/jcd.21375 fatcat:rfxf6fla7rfgdpoeawmatfzt74
., Hamiltonian decomposition of K,*, patterns with distinct differences, and Tuscan squares, Discrete Mathematics 91 (1991) 259-276. ... These Latin squares are called Row Complete Latin Squares  or Roman squares  . Most of the known Roman squares were obtained by polygonal path constructions [9, 11, 22] . ... A Latin square of order n is an n x n Italian square such that each column is a permutation of V,. ...doi:10.1016/0012-365x(90)90235-a fatcat:jipqsbrajrgrhpy4wlci5dqndu
The idea is to use a Hamiltonian double Latin square X to define the top n 2 layers of A. ... Let A, B be two order n 2 Latin squares and let σ ∈ S n 2 be a cyclic permutation. Then the block matrix X = A B σ(A) B . is an order n Hamiltonian double Latin square. ...doi:10.1007/s00454-013-9554-5 fatcat:vx7xa7zuxbarlblyy42opifojm
Preece, How many 7 x 7 Latin squares can be partitioned into Youden squares? (343- 352); D. A. Preece and B. J. ... Vowden, Graeco-Latin squares with embedded balanced superimpositions of Youden squares (353- 363); Zdenék Ryjaéek and Ingo Schiermeyer, On the independence number in K),,,\-free graphs (365-374); Abdén ...
A well-known conjecture of Thomassen says that every 4-connected line graph is hamiltonian. In this paper we prove that every 7-connected line graph is hamiltonian-connected. Michael 0. ... The following papers will be published in future issues: Siming Zhan, On hamiltonian line graphs and connectivity. ... Lamken, The existence of 3 orthogonal partitioned incomplete Latin squares of type P. ...doi:10.1016/0012-365x(90)90069-t fatcat:wzbr5q4ikzhxxb4lvuteckukqm
The author uses the construction of a row complete Latin square to show that Kj,, can be decomposed into 2m edge-disjoint directed Hamiltonian paths. ... A Latin square L is said to be row complete if, for all ordered pairs (u,v) of distinct symbols used in L, there is a row containing u and v as successive elements in the order given. ...
Suppose that H is the unique graph that satisfies the “Latin square graph” characterization but is not a Latin square graph. ... Moon [ibid. 34 (1963), 664- 667; MR 26 #5559] have characterized Latin square graphs. With one exception, a graph is a Latin square graph if and only if it has n? ...
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