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Quarter-regular biembeddings of Latin squares

2010
*
Discrete Mathematics
*

We apply a recursive construction for

doi:10.1016/j.disc.2009.08.020
fatcat:yetfifg7avhirjfalkiohrj7xm
*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. ...##
###
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.” ...##
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On the bigenus of the complete graphs
[article]

2021
*
arXiv
*
pre-print

These so-called

arXiv:2111.11671v1
fatcat:t4mcmbgg65h7xkztpm3vdqs7wq
*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. ...##
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On the number of non-isomorphic (simple) k-gonal biembeddings of complete multipartite graphs
[article]

2022
*
arXiv
*
pre-print

Moreover about the embeddings

arXiv:2111.08323v2
fatcat:tqyfb4pb45ehrcv63pibjjbtjq
*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 ...##
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On Biembeddings of Latin Squares

2009
*
Electronic Journal of Combinatorics
*

In turn, these

doi:10.37236/195
fatcat:gm7vlxuqq5fkzneq3ipggzk4fe
*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). ...##
###
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. ...##
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On λ-fold relative Heffter arrays and biembedding multigraphs on surfaces
[article]

2020
*
arXiv
*
pre-print

In the last part

arXiv:2010.10948v1
fatcat:dclss3brzrfzdmz3feivudc3ye
*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. ...##
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On Parity Vectors of Latin Squares

2010
*
Graphs and Combinatorics
*

Finally, we give a lower bound on the

doi:10.1007/s00373-010-0942-9
fatcat:nmcvdn55crcpllqaf2xzizc2ua
*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] . ...##
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Orientable biembeddings of cyclic Steiner triple systems from current assignments on Möbius ladder graphs

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 ...

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

2022
*
arXiv
*
pre-print

Moreover, in case t=1 and v is a prime, almost all these embeddings define faces that are all

arXiv:2205.02066v1
fatcat:yskwh4w3zffylkehobkybbiflu
*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 . ...##
###
Triangulations of the sphere, bitrades and abelian groups
[article]

2013
*
arXiv
*
pre-print

Connections are made between the structure

arXiv:1112.5423v3
fatcat:vuaigsjwvzgm5bvpmfbkq2wmpu
*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? ...##
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Infinite families of bi-embeddings

1990
*
Discrete Mathematics
*

Several

doi:10.1016/0012-365x(90)90320-h
fatcat:yoqsuurmf5g5hhd225yo4u5uc4
*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 ...##
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THIRD-REGULAR BI-EMBEDDINGS OF LATIN SQUARES

2010
*
Glasgow Mathematical Journal
*

The corresponding Latin squares

doi:10.1017/s0017089510000376
fatcat:lmxzeu4w6zfcvbvgp6ocov4z6y
*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 ...##
###
Combinatorial Embeddings and Representations

2012

Finally, we provide an enumeration

doi:10.21954/ou.ro.0000f1d5
fatcat:pxofnx53jrcehflgkdv5qi66v4
*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). ...##
###
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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