A Note on Quadrangular Embedding of Abelian Cayley Graphs
DOI:
https://doi.org/10.5540/tema.2016.017.03.0331Keywords:
Abelian Cayley Graphs, Genus of a graph, Flat torus, Tessellations.Abstract
The genus graphs have been studied by many authors, but just a few results concerning in special cases: Planar, Toroidal, Complete, Bipartite and Cartesian Product of Bipartite. We present here a general lower bound for the genus of a abelian Cayley graph and construct a family of circulant graphs which reach this bound.
References
A. Ádám. Research problem 2-10. J. Combinatorial Theory, (1967).
B. Alspach & T.D. Parsons. Isomorphism of circulant graphs and digraphs. Discrete Math., 25(2) (1979), 97–108.
F. Boesch & R. Tindell. Circulants and their connectivities. J. Graph Theory, 8(4) (1984), 487–499.
L. W. Beineke & F. Harary. The Genus of the n-Cube. Canad. J. Math., 17 (1965), 494–496.
M. Conder & R. Grande. On Embeddings of Circulant Graphs. The Eletronic Journal of Combinatorics, 22(2) (2015), 1–27.
S.I.R Costa, J.E Strapasson, M.Muniz & T.B. Carlos. Circulant graphs and tessellations on flat tori. Linear Algebra and Appl, 432(1) (2010), 369–382.
J.E. Gross & T.W. Tucker. Topological graph theory. Dover Publications Inc., Mineola, NY (2001).
F. Harary, Graph Theory. Reading, MA: Addison-Wesley, (1994).
C. Heuberger. On planarity and colorability of circulant graphs. Discrete Math., 268(1-3) (2003), 153–169.
X. Lin, Y. Yang, J. Lü & X. Hao. The crossing number of C(mk;{1,k}). Graphs Combin., 21(1) (2005), 89–96.
V. Liskovets & R. Pöschel. Counting circulant graphs of prime power order by decomposing into orbit enumeration problems. Discrete Math., 214(1-3) (2000), 173–191.
D. Matthew. On Hamilton cycle decomposition of 6-regular circulant graphs. Graphs Combin., 22(3) (2006), 331–340.
F.P. Muga II. Undirected Circulant Graphs, International Symposium on Parallel Architectures. Algorithms and Networks, (1994), 113–118.
J.J. Molitierno. On the algebraic connectivity of graphs as a function of genus. Linear Algebra and its Applications, 419(2-3) (2006), 519–531.
M. Muzychuk. Ádám’s conjecture is true in the square-free case. J. Combin. Theory Ser. A, 72(1) (1995), 118–134.
T. Pisanski. Genus of Cartesian products of regular bipartite graphs. J. Graph Theory, 4(1980), 31–42.
G. Ringel. Das Geschlecht des vollständiger Paaren Graphen. Abh. Math. Sem. Univ. Hamburg, 28 (1965), 139–150.
G. Ringel, Der vollständige paare Graph auf nichtorientierbaren Flächen. J. Reine Angew. Math., 220 (1965), 88–93.
G. Ringel, Über drei kombinatorische Problem am n-dimensionalen Würfel und Wurfelgitter. Abh. Math. Sem. Univ. Hamburg, 20 (1965), 10–19.
G. Ringel & J.W.T. Youngs. Solution of the Heawood Map-Coloring Problem. Proc. Nat. Acad. Sci. USA, 60 (1968), 438–445.
R.J. Trudeau. Introduction to graph theory. Dover Publications Inc., New York, 1993. Corrected reprint of the 1976 original.
A.T. White. The genus of repeated cartesian products of bipartite graphs. Trans. Amer. Math. Soc., 151 (1970), 393–404.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.