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Orientable quadrilateral embeddings of products of graphs

Tomaž Pisanski
1992 Discrete Mathematics  
., Orientable 109 (1992) 203-205. quadrilateral embeddings of products of graphs, Discrete Mathematics It is proved that for each connected regular graph G there exists an integer m > 0 such that for each  ...  integer n am the Cartesian product G x Q, has an orientable quadrilateral embedding, thereby extending the result from bipartite to regular nonbipartite graphs.  ...  White for careful reading of the manuscript and for helpful hints and suggestions. As a matter of fact recent joint work in progress with A.T.  ... 
doi:10.1016/0012-365x(92)90291-m fatcat:5oexlxqlc5hw7jajz4tlm2yxri

Nonorientable genus of nearly complete bipartite graphs

Bojan Mohar
1988 Discrete & Computational Geometry  
In particular, it is shown that G(5,4,4) and G(5, 5, 5) have no nonorientable quadrilateral embedding.  ...  Let G(m, n, k), m, n > 3, k-< min(m, n), be the graph obtained from the complete bipartite graph K,.. by deleting an arbitrary set of k independent edges, and let It is shown that the nonorientable genus  ...  On the other hand, genus embeddings of some more "general" bipartite graphs with many edges can also be obtained using similar methods as in this paper.  ... 
doi:10.1007/bf02187903 fatcat:otdkgt36gve3hk5axnq24fz4iq

Page 3797 of Mathematical Reviews Vol. , Issue 85i [page]

1985 Mathematical Reviews  
The techniques in this paper are used to show that in addition, quadrilateral embedding of a bipartite graph G with at least 3 vertices can be lifted to a quadrilateral embedding of G[K,,| for every odd  ...  D. (1-PAS) Embeddings of bipartite graphs. J. Graph Theory 7 (1983), no. 3, 325-334.  ... 

Nonorientable Embeddings of Groups

Tomaž Pisanski, Arthur T. White
1988 European journal of combinatorics (Print)  
Embeddings of Cayley graphs into nonorientable surfaces are studied. Some lower and upper bounds for the nonorientable genus of abelian and hamiltonian groups are obtained.  ...  In some cases the method is applied to orientable embeddings.  ...  ACKNOWLEDGEMENTS The authors acknowledge the help of Chris Godsil with the proof of Proposition 5.  ... 
doi:10.1016/s0195-6698(88)80003-9 fatcat:3afbn2lst5hwzgga2fpo3nlhtq

Graph-Theoretic Solutions to Computational Geometry Problems [chapter]

David Eppstein
2010 Lecture Notes in Computer Science  
, all interior faces quadrilaterals, all interior vertex degrees ≥ 4 (bipartite, median graph) Kinggraph: add diagonals of quads Guaranteed to be 4-chromatic 4-color, use smallest color class  ...  having pq as their diameter = max ind. set of bipartite graph Dynamic graph algorithm for multiple bipartite MIS's [E. & Erickson 1994] To find max cluster size for given diameter D: For each input  ...  Embedding: 1-1 function from the points of one metric space to another dilation of a single distance: distortion of embedding = worst ratio of dilations if embedding scaled so all distances nondecreasing  ... 
doi:10.1007/978-3-642-11409-0_1 fatcat:rqrljmkgqrb3ddw5nszva6ighi

Dimers and Circle patterns [article]

Richard Kenyon, Wai Yeung Lam, Sanjay Ramassamy, Marianna Russkikh
2020 arXiv   pre-print
We establish a correspondence between the dimer model on a bipartite graph and a circle pattern with the combinatorics of that graph, which holds for graphs that are either planar or embedded on the torus  ...  The set of positive face weights on the graph gives a set of global coordinates on the space of circle patterns with embedded dual.  ...  Acknowledgements We are grateful to Dmitry Chelkak for sharing his ideas during many fruitful exchanges, and providing the proof of Lemma 11. R. Kenyon  ... 
arXiv:1810.05616v2 fatcat:3hnnbhodgjbdxoscnbfow3kwqq

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

1997 Mathematical Reviews  
Summary: “Let G be a bipartite graph with a quadrilateral (or self-dual) embedding.  ...  (SAR-UPAM; Dhahran) Self-dual embeddings of the composition of bipartite graphs. (English summary) Graph theory, combinatorics, and algorithms, Vol. \, 2 (Kalamazoo, MI, 1992), 1-4, Wiley-Intersci.  ... 

The construction and classification of self-dual spherical polyhedra

Dan Archdeacon, R Bruce Richter
1992 Journal of combinatorial theory. Series B (Print)  
In this paper we consider spherical polyhedra, or equivalently 3-connected embedded planar graphs. A self-duality map sends vertices to faces and faces to vertices while preserving incidence.  ...  Included is the construction of polyhedra which admit only self-duality maps of large order. (i3  ...  It follows that R -E(A,) is an embedded bipartite graph with each face a quadrilateral or a hexagon.  ... 
doi:10.1016/0095-8956(92)90065-6 fatcat:vzhcz3ljvvcfljahzpt7toja7u

On the genus of the tensor product of graphs where one factor is a regular graph

Tamara Dakić, Tomaž Pisanski
1994 Discrete Mathematics  
Assuming that H is suitably imbedded in some orientable surface by modifying the edges of H according to the configuration of G we get the permutation voltage graph H c2') whose permutation derived graph  ...  The tensor product HOG where G is a 2k-regular graph can be regarded as a covering space of the permutation voltage graph H czk) obtained from H.  ...  where H is a bipartite (p, q) graph which has an orientable quadrilateral imbedding.  ... 
doi:10.1016/0012-365x(93)e0057-b fatcat:7ehxteqeanayxex6q5jinfa3am

The largest demigenus of a bipartite signed graph

Thomas Zaslavsky
2001 Discrete Mathematics  
We ÿnd the smallest surface within which it is possible to orientation-embed the complete bipartite signed graph ±Kr;s, which is obtained from the complete bipartite graph Kr;s through replacing each edge  ...  A graph with signed edges is orientation embedded in a surface when it is topologically embedded so that one trip around a closed path preserves or reverses orientation according as the path's sign product  ...  Quadrilaterality. Since a minimal embedding of K 2r; 2s has only quadrilateral faces, the same holds true for a minimal embedding of ±K r; s .  ... 
doi:10.1016/s0012-365x(00)00349-6 fatcat:rlrg7wqfhvafhgmrvvbbmjntaq

Page 5880 of Mathematical Reviews Vol. , Issue 93k [page]

1993 Mathematical Reviews  
Giorgio Balconi (I-PAVI-G) COMBINATORICS 93k:05053 05C10 Pisanski, Tomaz (SV-LJUB) Orientable quadrilateral embeddings of products of graphs.  ...  The main result of this paper is that a connected projective-planar graph G which is not planar has a projective-planar embedding with only even faces if and only if either G is bipartite or B(G) is planar  ... 

Tight embeddings of partial quadrilateral packings

Zoltán Füredi, Jenő Lehel
2010 Journal of combinatorial theory. Series A  
Include the quadrilaterals (a 1 , a 2 , a 3 , b 1 ), (a 3 , a 4 , a 5 , b 2 ), (b 1 , a 2 , b 2 , a 4 ), and add a C 4 -decomposition of the complete bipartite graph K 2,n−5 (see (2) ) with parts B and  ...  Remove a maximal set of edge disjoint quadrilaterals from E(G 1 ) and include them to P. The remaining graph of the uncovered edges, G 2 , is quadrilateral-free.  ... 
doi:10.1016/j.jcta.2009.06.003 fatcat:wt7fbz6nljhnrj7eziuhtnciqa

Page 1673 of Mathematical Reviews Vol. , Issue 2000c [page]

2000 Mathematical Reviews  
We study several problems related to such bipartite embeddings.” 2000c:05048 2000c:05045 05C10 Abu-Sbeih, Moh’d Z. (SAR-UPAM; Dhahran) Diagonalizable embedding of composition graphs.  ...  (E-UPMAT; Madrid) Bipartite embeddings of trees in the plane. (English summary) Discrete Appl. Math. 93 (1999), no. 2-3, 141-148. Summary: “In this paper we consider the following embedding problem.  ... 

Self-dual embeddings of complete bipartite graphs

Dan Archdeacon, Nora Hartsfield
1992 Journal of combinatorial theory. Series B (Print)  
In this paper we examine self-dual embeddings of the complete bipartite graph K", on both orientable and nonorientable surfaces.  ...  We show by construction that these necessary conditions are sufficient, except that there is no orientable self-dual embedding of K6,6. 0  ...  ACKNOWLEDGMENT The authors thank Richard Wilson for the embedding of Kp dual to K6,6  ... 
doi:10.1016/0095-8956(92)90056-4 fatcat:6mnkxinrejhgrcc3gys4z3ja64

Page 45 of Mathematical Reviews Vol. , Issue 92a [page]

1992 Mathematical Reviews  
A quadrilateral embedding of a graph G with 2n vertices in a closed surface S is called diagonalizable if there exists a set of n quadrilaterals of the embedding and a choice of diagonal in each of these  ...  It seems that determining whether an arbitrary graph G admits a diagonalizable quadrilateral embedding is a difficult problem.  ... 
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