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Quadrangular embeddings of the complete even k-partite graph

Nora Hartsfield, Gerhard Ringel
1990 Discrete Mathematics  
The complete even k-partite graph K n,.n* ,..., "* is the complete k-partite graph where all the n,'s are even numbers.  ...  Orientable and nonorientable quadrangular embeddings are constructed for all these graphs.  ...  Each complete even k-partite graph has a quadrangular embedding into a nonorientable 2-manifold.  ... 
doi:10.1016/0012-365x(90)90175-h fatcat:u3j2bjenajgilawcyzri64oohm

Minimal quadrangulations of orientable surfaces

Nora Hartsfield, Gerhard Ringel
1989 Journal of combinatorial theory. Series B (Print)  
Quadrangular embeddings of the general octahedral graph and the complete graph K" for n = 5 (mod 8) into orientable surfaces are constructed.  ...  These embeddings determine polyhedra with a minimal number of quadrangles.  ...  An alternative description is that 02,, is the complete n-partite graph with two vertices in each partite set. Figure 10 shows a quadrangular embedding of 0s into S,.  ... 
doi:10.1016/0095-8956(89)90009-9 fatcat:cbiyzzlm65evpndwg2gatfdfri

Page 1292 of Mathematical Reviews Vol. , Issue 91C [page]

1991 Mathematical Reviews  
It is a complete even k-partite graph if the cardinality of each V; is even.  ...  The authors show that each complete even k-partite graph has a quadrangular embedding in an orientable 2-manifold and in a nonorientable 2- manifold. M. L.  ... 

Minimal quadrangulations of nonorientable surfaces

Nora Hartsfield, Gerhard Ringel
1989 Journal of combinatorial theory. Series A  
An alternative description is that OZn is the complete n-partite graph with two vertices in each partite set.  ...  QUADRANGULAR EMBEDDINGS OF K, FOR n= l(mod4) The following is the scheme for a quadrangular embedding of K,, into a nonorientable surface of genus 28: 0. 1, 9, 11, 4, 5, 10, 8, 6, 12, 7, 2, 3 1. 2, 10,  ... 
doi:10.1016/0097-3165(89)90014-9 fatcat:hpzpgbsex5frdpghlim3f7mufy

Author index Volume 81 (1990)

1990 Discrete Mathematics  
Ringel, Quadrangular embeddings of the complete even k-partite graph Hertz, A., Bipolarizable graphs A.M., On the independence number of random graphs (2) 171-175 Guha, S. and S.  ...  Padmanabhan, A new derivation of the generating function for the major index (Communicution) Guichard, D.R., Two theorems on the addition of residue classes Gyarfas, A., see Chung, F.R.K.  ... 
doi:10.1016/0012-365x(90)90070-x fatcat:p3edatrggzavxcniotek3dqgv4

Genus of the Cartesian Product of Triangles

Michal Kotrbčík, Tomaž Pisanski
2015 Electronic Journal of Combinatorics  
The (strong) symmetric genus range of irredundant Cayley graphs of $Z_p^n$ is calculated for all odd primes $p$.  ...  Using a lifting method we present a general construction of a low-genus embedding of $G_n$ using a low-genus embedding of $G_{n-1}$.  ...  Part of the research was done while the first author was visiting the Department of Mathematics of the University of Ljubljana and he would like to thank the host and the department for hospitality.  ... 
doi:10.37236/2951 fatcat:wismx3w5abeuxdsgmwum4thqu4

A Basic Structure for Grids in Surfaces [article]

Lowell Abrams, Daniel Slilaty
2019 arXiv   pre-print
A graph G embedded in a surface S is called an S-grid when every facial boundary walk has length four, that is, the topological dual graph of G in S is 4-regular.  ...  with the S-grid G is unique and so our characterization of S-grids also partitions the collection of all S-grids.  ...  If G is graph embedded in a surface S, then the edges of G partition in exactly one way into transverse walks and transverse circuits.  ... 
arXiv:1901.03682v1 fatcat:j4zenx3novfqdhmm44kro4kn2u

Page 2340 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
In addition, we give the crossing number of the graph obtained by adding two edges to the graph K;4,, in such a way that these new edges join a vertex of degree n + | of the graph K,,4,, with two of its  ...  By means of a large number of inequivalent single-face embeddings of index one current graphs the authors prove that, for each s >1, the complete graphs Kj2,,4 and Kj)2,,7 have at least 4° non-isomorphic  ... 

Hamilton surfaces for the complete even symmetric bipartite graph

Nora Hartsfield, Brad Jackson, Gerhard Ringel
1989 Discrete Mathematics  
The authors discuss a hamilton surface of a graph, which is a two-dimensional analog of a hamilton cycle.  ...  Cl Hamilton surfaces for symmetric bipartite graph Somp related questions We considered the problem of obtaining hamilton surface decompositions 93 of the complete graph K,.  ...  These polyhedra are in fact genus embeddings of the graph consisting of the vertices and edges of Qm.  ... 
doi:10.1016/0012-365x(89)90164-7 fatcat:x7jgdacoxjdave4mxa7nbwhspu

Page 6451 of Mathematical Reviews Vol. , Issue 96k [page]

1996 Mathematical Reviews  
Summary: “Nonorientable quadrangular embeddings are con- structed for almost all complete multipartite graphs where V — 'E is an integer.  ...  Steve Fisk (1-BOWD; Brunswick, ME) 96k:05042 05C10 Hartsfield, Nora (1-WWA; Bellingham, WA) Nonorientable quadrangular embeddings of complete multipartite graphs.  ... 

Hamiltonian cycles in bipartite toroidal graphs with a partite set of degree four vertices

Jun Fujisawa, Atsuhiro Nakamoto, Kenta Ozeki
2013 Journal of combinatorial theory. Series B (Print)  
Let G be a 3-connected bipartite graph with partite sets X ∪ Y which is embeddable in the torus.  ...  In order to prove the result, we establish a result on orientations of quadrangular torus maps possibly with multiple edges.  ...  Acknowledgments The authors are grateful to two anonymous referees for their careful reading of the paper and helpful suggestions for improving the presentation.  ... 
doi:10.1016/j.jctb.2012.08.004 fatcat:644jj5vqaffrpgqlh4xxntmnji

Non-crossing trees, quadrangular dissections, ternary trees, and duality preserving bijections [article]

Nikos Apostolakis
2020 arXiv   pre-print
Using the theory of Properly Embedded Graphs developed in an earlier work we define an involutory duality on the set labeled non-crossing trees that lifts the obvious duality in the set of unlabeled non-crossing  ...  In particular the bijection from the set of quadrangular dissections to the set of non-crossing trees seems to be new.  ...  A Properly Embedded Graph (peg for short) is a graph embedded in a compact oriented surface with boundary in such a way that: • the vertices of the graph lie on the boundary of the surface and the interior  ... 
arXiv:1807.11602v2 fatcat:42iizx7scvgd3fivy3a6ti5q6m

Efficient polynomial-time approximation scheme for the genus of dense graphs [article]

Yifan Jing, Bojan Mohar
2020 arXiv   pre-print
minimum genus embedding in the sense that the minimum genus 𝗀(G) of G satisfies: 𝗀(G)≤ g≤ (1+ε)𝗀(G).  ...  The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs.  ...  Mohar [31] proved that the genus is NP-complete even if we restrict our attention to apex graphs, i.e. graphs which become planar by removing a single vertex.  ... 
arXiv:2011.08049v1 fatcat:bujrvl2nnrg3ja7mtzu5retadm

The 2-extendability of 5-connected graphs on the Klein bottle

Seiya Negami, Yusuke Suzuki
2010 Discrete Mathematics  
More generally, Plummer [7] has shown that for a given closed surface, there exists an upper bound for a natural number k such that the surface admits embeddings of k-extendable graphs and Dean [3] has  ...  A graph is said to be k-extendable if any independent set of k edges extends to a perfect matching.  ...  For our purpose, we first embed a complete bipartite graph K 3,3 into the Klein bottle K 2 , assuming that two partite sets of the K 3,3 are S = {s 1 , s 2 , s 3 } and X = {x 1 , x 2 , x 3 }.  ... 
doi:10.1016/j.disc.2010.06.020 fatcat:nojktoeqnjfj7mjd5umtqxot5y

On the genus of the complete tripartite graph Kn,n,1

Valentas Kurauskas
2017 Discrete Mathematics  
For even n we prove that the genus of the complete tripartite graph K_n,n,1 is (n-1) (n-2)/4 .  ...  This is the least number of bridges needed to build a complete n-way road interchange where changing lanes is not allowed.  ...  I would like to sincerely thank Rimvydas Krasauskas who introduced me to the road interchange problem and suggested to model it via graph embeddings.  ... 
doi:10.1016/j.disc.2016.09.017 fatcat:tmooydhzrjbgxnt4ug743eecuu
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