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Quarter-regular biembeddings of Latin squares

D.M. Donovan, A. Drápal, M.J. Grannell, T.S. Griggs, J.G. Lefevre
2010 Discrete Mathematics  
We apply a recursive construction for biembeddings of Latin squares to produce a new infinite family of biembeddings of cyclic Latin squares of even side having a high degree of symmetry.  ...  Reapplication of the construction yields two further classes of biembeddings.  ...  Acknowledgements The authors would like to acknowledge the support of the Australian Research Council, grant numbers LX0453416 and DP0664030, and the Australian Mathematical Sciences Institute.  ... 
doi:10.1016/j.disc.2009.08.020 fatcat:yetfifg7avhirjfalkiohrj7xm

Page 2114 of Mathematical Reviews Vol. , Issue 97D [page]

1997 Mathematical Reviews  
A biembedding is called perfect if g; = q2. An infinite family of perfect biembeddings of complete graphs is exhibited for nonori- entable surfaces.  ...  Several additional infinite families of nonori- entable biembeddings are shown as well as biembeddings on one orientable and one nonorientable surface. Finally, some unsolved problems are presented.”  ... 

On the bigenus of the complete graphs [article]

Timothy Sun
2021 arXiv   pre-print
These so-called biembeddings solve a generalization of the Earth-Moon problem for an infinite number of orientable surfaces.  ...  We describe an infinite family of edge-decompositions of complete graphs into two graphs, each of which triangulate the same orientable surface.  ...  at infinitely many values.  ... 
arXiv:2111.11671v1 fatcat:t4mcmbgg65h7xkztpm3vdqs7wq

On the number of non-isomorphic (simple) k-gonal biembeddings of complete multipartite graphs [article]

Simone Costa, Anita Pasotti
2022 arXiv   pre-print
Moreover about the embeddings of K_v/t× t, for t∈{1,2,k}, we provide a construction of 2^v·H(1/4)/2k(k-1)+o(v,k) non-isomorphic k-gonal biembeddings whenever k is odd and v belongs to a wide infinite family  ...  This article aims to provide exponential lower bounds on the number of non-isomorphic k-gonal biembeddings of the complete multipartite graph into orientable surfaces.  ...  In particular, in Section 4 we obtain that, when k is congruent to 3 modulo 4 and v belongs to an infinite family of values, there are k k 2 +o(k) •2 g(k)v+o(v) non-isomorphic k-gonal biembeddings of K  ... 
arXiv:2111.08323v2 fatcat:tqyfb4pb45ehrcv63pibjjbtjq

On Biembeddings of Latin Squares

M. J. Grannell, T. S. Griggs, M. Knor
2009 Electronic Journal of Combinatorics  
In turn, these biembeddings enable us to increase the best known lower bound for the number of face 2-colourable triangular embeddings of $K_{n,n,n}$ for an infinite class of values of $n$.  ...  This facilitates biembeddings of a wide variety of Latin squares, including those formed from the Cayley tables of the elementary Abelian 2-groups $C_2^k$ ($k\ne 2$).  ...  However, up until now, the only clearly identifiable family of Latin squares that are known to admit biembeddings has been the family of cyclic squares C n defined by C n (i, j) = i + j (mod n).  ... 
doi:10.37236/195 fatcat:gm7vlxuqq5fkzneq3ipggzk4fe

Page 429 of Mathematical Reviews Vol. , Issue 81B [page]

1981 Mathematical Reviews  
Anderson, I. 81b:05034 Infinite families of biembedding numbers. J. Graph Theory 3 (1979), no. 3, 263-268.  ...  Well-known embedding theorems are used to establish several infinite families of values of N(y, y’); for exam- ple: For all n> 10, N((n— 1)’, 4{(n — 3)(n— 4)/12}) =4n+1.  ... 

On λ-fold relative Heffter arrays and biembedding multigraphs on surfaces [article]

Simone Costa, Anita Pasotti
2020 arXiv   pre-print
In the last part of the paper we also show that these arrays give rise to biembeddings of multigraphs into orientable surfaces and we provide infinite families of such biembeddings.  ...  we present existence results for infinite classes of these arrays.  ...  Some topological considerations In this paper we have introduced the concept of relative Heffter arrays and we have provided constructions for infinite families of such objects.  ... 
arXiv:2010.10948v1 fatcat:dclss3brzrfzdmz3feivudc3ye

On Parity Vectors of Latin Squares

D. M. Donovan, M. J. Grannell, T. S. Griggs, J. G. Lefevre
2010 Graphs and Combinatorics  
Finally, we give a lower bound on the number of main classes of Latin squares of side n that admit no self-embeddings.  ...  vectors of the constituent factors.  ...  In conclusion, we note that the number of nonisomorphic biembeddings of Latin squares of side n is known to be at least n n 2 144 (1−o(1)) for an infinite set of values of n [4] .  ... 
doi:10.1007/s00373-010-0942-9 fatcat:nmcvdn55crcpllqaf2xzizc2ua

Orientable biembeddings of cyclic Steiner triple systems from current assignments on Möbius ladder graphs

M.J. Grannell, V.P. Korzhik
2009 Discrete Mathematics  
biembedding of two cyclic Steiner triple systems of order 12n + 7.  ...  These produce many nonisomorphic orientable biembeddings of cyclic Steiner triple systems of order 12n + 7.  ...  We prove in Theorem 4 that, for a certain infinite set of values of n, there is a positive constant B such that the number of skew Skolem sequences of order n is at least Bn 1 log 2 9 , although we suspect  ... 
doi:10.1016/j.disc.2008.07.016 fatcat:44y553pdjza7bj3kpz7mvb7asm

Biembeddings of Archdeacon type: their full automorphism group and their number [article]

Simone Costa
2022 arXiv   pre-print
Moreover, in case t=1 and v is a prime, almost all these embeddings define faces that are all of the same length kv, i.e. we have a more than exponential number of non-isomorphic kv-gonal biembeddings  ...  As an application of this result, given a positive integer t≢04, we prove that there are, for infinitely many pairs of v and k, at least (1-o(1)) (v-t/2)!  ...  array), i.e. we have a more than an exponential number of non-isomorphic kv-gonal biembeddings of K v .  ... 
arXiv:2205.02066v1 fatcat:yskwh4w3zffylkehobkybbiflu

Triangulations of the sphere, bitrades and abelian groups [article]

Simon R. Blackburn, Thomas A. McCourt
2013 arXiv   pre-print
Connections are made between the structure of A_W and the theory of asymmetric Laplacians of finite directed graphs, and weaker results for orientable surfaces of higher genus are given.  ...  The relevance of the group A_W to the understanding of the embeddings of a partial latin square in an abelian group is also explained.  ...  Is there a family of bitrades where the minimal rank of an abelian group in which both mates embed is linear in the size of the bitrade?  ... 
arXiv:1112.5423v3 fatcat:vuaigsjwvzgm5bvpmfbkq2wmpu

Infinite families of bi-embeddings

Sharon Cabaniss, Bradley W. Jackson
1990 Discrete Mathematics  
Several infinite families of bi-embeddings of complete graphs are exhibited. Results from a computer search for additional bi-embeddings are also included. Finally, an unsolved problem is presented.  ...  of genera p, and p2 respectively.  ...  In [3] Anderson and White found seven new cases using current graphs, and in [13] Anderson found some infinite families of bi-embeddings using previously known embeddings of K scnj and SK,, for s = 3  ... 
doi:10.1016/0012-365x(90)90320-h fatcat:yoqsuurmf5g5hhd225yo4u5uc4


2010 Glasgow Mathematical Journal  
The corresponding Latin squares of side n are determined, together with the full automorphism group of the embedding.  ...  The current paper presents an analysis of the face 2-colourable triangular embedding of K n,n,n that results from this.  ...  It is perhaps appropriate at this point to note that in a recent paper [3] , a new infinite family of face 2-colourable triangular embeddings of complete tripartite graphs K n,n,n was constructed having  ... 
doi:10.1017/s0017089510000376 fatcat:lmxzeu4w6zfcvbvgp6ocov4z6y

Combinatorial Embeddings and Representations

Constantinos Psomas
Finally, we provide an enumeration of graphs of up to six edges representable by Steiner triple systems.  ...  Topological embeddings of complete graphs and complete multipartite graphs give rise to combinatorial designs when the faces of the embeddings are triangles.  ...  In such a biembedding, the number of vertices, V = 3n, the number of edges, E = 3(n2 -n),-and the number of faces, F = 2(n2 -n).  ... 
doi:10.21954/ fatcat:pxofnx53jrcehflgkdv5qi66v4

Page 19 of Mathematical Reviews Vol. , Issue Annual Index [page]

Mathematical Reviews  
Infinite families of biembedding numbers. J. bar EY 263-268. (A. T. White) 81b:05034 05C10 Anderson, J. M. On a theorem of Rudin concerning the class A. J. London Math.  ...  Calculation of internal viscous flows in ic ducts at moderate to, high Reynolds numbers. Comput. & Fluids 8 (1980), no. 4, 391-411. (Author’s summary) 81g:76034 76D05 Anderson, Oliver D.  ... 
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