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On the non-orientable genus of a 2-connected graph

R.Bruce Richter
<span title="">1987</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
Analogously define the genus and non-orientable genus of G. This paper is devoted to the proof of the following "additivity" result. THEOREM 1.  ...  To ensure that 2 is non-orientable, observe that 0, +ez < 2 implies that we can take at least one of Z', and C, to be non-orientable. 1 Before we can prove Lemma 3, we need a result (Theorem 2) which completely  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0095-8956(87)90029-3">doi:10.1016/0095-8956(87)90029-3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ug6w23dm7jhmfisg3n7wflwxuy">fatcat:ug6w23dm7jhmfisg3n7wflwxuy</a> </span>
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Compact Klein surfaces of genus $5$ with a unique extremal disc

Gou Nakamura
<span title="2013-02-28">2013</span> <i title="American Mathematical Society (AMS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/k76rcy45xne6jlpmdtcz26fvze" style="color: black;">Conformal Geometry and Dynamics</a> </i> &nbsp;
The present paper concerns non-orientable extremal surfaces of genus 5.  ...  A compact (orientable or non-orientable) surface of genus g is said to be extremal if it contains an extremal disc, that is, a disc of the largest radius determined only by g.  ...  Figure 2 shows a trivalent graph, a closed walk on it, and a sidepairing pattern, where a line (resp. a dotted line) connecting two sides of the regular 24-gon denotes a pair with the opposite (resp.  ... 
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Duke's theorem does not extend to signed graph embeddings

Jozef Širáň
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Using homology-type arguments and surface surgery it is proved that a direct extension of the classical Duke's contiguity theorem to cellular orientation embeddings of signed graphs is impossible.  ...  ., Duke's theorem does not extend to signed graph embeddings, Discrete Mathematics, 94 (1991) 233-238.  ...  Acknowledgements Thanks are due to A. Bouchet and M. Skoviera for valuable discussions on the topic.  ... 
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Compact 3-manifolds via 4-colored graphs

Paola Cristofori, Michele Mulazzani
<span title="2015-07-24">2015</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/qgwiiycdsff5niydy5k3styeby" style="color: black;">RACSAM</a> </i> &nbsp;
classification of compact 3-manifolds representable by graphs with few vertices (< 6 in the non-orientable case and < 8 in the orientable one).  ...  Our construction is a direct generalization of the one given in the eighties by S.  ...  Γ ′ 1 ) is a 4-colored graph representing the genus one orientable (resp. non-orientable) handlebody.  ... 
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Strong embeddings of minimum genus

Bojan Mohar
<span title="">2010</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
Waterloo, 1977), pp. 341-355, Academic Press, 1979]) asserts that every bridgeless cubic graph can be embedded on a surface of its own genus in such a way that the face boundaries are cycles of the graph  ...  In this paper we consider closed 2-cell embeddings of graphs and show that certain (cubic) graphs (of any fixed genus) have closed 2-cell embedding only in surfaces whose genus is very large (proportional  ...  If there is a 2-edge-connected graph whose genus (or non-orientable genus) is at most g and that does not have a cycle double cover, then there is such a graph G with the following properties: (a) G is  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2010.03.019">doi:10.1016/j.disc.2010.03.019</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6gctsgfi65etrkn4lms2s6r53e">fatcat:6gctsgfi65etrkn4lms2s6r53e</a> </span>
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Combinatorial local planarity and the width of graph embeddings

Bojan Mohar
<span title="1992-12-01">1992</span> <i title="Canadian Mathematical Society"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/m5ds6a3cpng5df7duilz7mzivi" style="color: black;">Canadian Journal of Mathematics - Journal Canadien de Mathematiques</a> </i> &nbsp;
The criterion is independent of embeddings of the graph, but it guarantees that a given cycle in a graph G must be contractible in any minimal genus embedding of G (either orientable, or non-orientable  ...  Let G be a graph embedded in a closed surface. The embedding is "locally planar" if for each face, a "large" neighbourhood of this face is simply connected.  ...  I am greatly indebted to Steve Fisk for a careful reading of the manuscript.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.4153/cjm-1992-076-8">doi:10.4153/cjm-1992-076-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gcvtl2auq5hrtljemdkcruz6vm">fatcat:gcvtl2auq5hrtljemdkcruz6vm</a> </span>
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On the matching extendability of graphs in surfaces

R.E.L. Aldred, Ken-ichi Kawarabayashi, Michael D. Plummer
<span title="">2008</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
many graphs of orientable genus g which are 3-extendable, and given g 2, there are infinitely many graphs of non-orientable genus g which are 3-extendable; and (3) if G is a 5-connected triangulation  ...  It is shown that (1) if a graph is embedded on a surface of Euler characteristic χ , and the number of vertices in G is large enough, the graph is not 4-extendable; (2) given g > 0, there are infinitely  ...  Acknowledgments The authors thank the anonymous referees for a thorough job and helpful suggestions leading to a substantial shortening of the proof of Theorem 2.1 as well as a simpler non-orientable construction  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jctb.2007.06.001">doi:10.1016/j.jctb.2007.06.001</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qwthekxgerglnniw3ajztf7zxm">fatcat:qwthekxgerglnniw3ajztf7zxm</a> </span>
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A Heawood-type result for the algebraic connectivity of graphs on surfaces [article]

Pedro Freitas
<span title="2001-09-24">2001</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We prove that the algebraic connectivity a(G) of a graph embedded on a nonplanar surface satisfies a Heawood-type result.  ...  As an application of these results and techniques, we obtain a lower bound for the genus of Ramanujan graphs.  ...  The orientable (non-orientable) genus of a graph is defined to be the smallest possible genus of an orientable (resp. non-orientable) surface where G is embeddable.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0109191v1">arXiv:math/0109191v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sldd75uwcjagtn3mdjj7kgkbku">fatcat:sldd75uwcjagtn3mdjj7kgkbku</a> </span>
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Characterization of signed graphs which are cellularly embeddable in no more than one surface

Jozef Širáň
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
We consider embeddings of signed graphs in which the balanced cycles of .he graph mducs orientation-preserving cycles on the surface. and characterize those signed graphs which are (in this sense) cellularly  ...  Characterization of signed graphs which are cellularly emheddahle in no more than one surface. Discrete Mathematics 94 (1991) 39-44.  ...  The non-orientable counterpart to this res& Lr Lt'nnected graphs G with at least one cycle follows at once from [4], For such G, the bmallest genus of a non-orientable surface into which G embeds cellularly  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(91)90304-k">doi:10.1016/0012-365x(91)90304-k</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/i43q5hrnhbayjiacvio6iv6fiy">fatcat:i43q5hrnhbayjiacvio6iv6fiy</a> </span>
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The bondage number of graphs on topological surfaces and Teschner's conjecture

Andrei Gagarin, Vadim Zverovich
<span title="">2013</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
genera of the graph, and show tight lower bounds for the number of vertices of graphs 2-cell embeddable on topological surfaces of a given genus.  ...  We provide constant upper bounds for the bondage number of graphs on topological surfaces, improve upper bounds for the bondage number in terms of the maximum vertex degree and the orientable and non-orientable  ...  Acknowledgement The authors are grateful to the anonymous referees for their valuable comments and remarks.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2012.12.018">doi:10.1016/j.disc.2012.12.018</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/y4foeftg6bgxlaq7ivjihhm53q">fatcat:y4foeftg6bgxlaq7ivjihhm53q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20140103104243/http://www.cems.uwe.ac.uk:80/~vzverovi/The%20bondage%20number%20of%20graphs%20on%20topological%20surfaces%20and%20Teschners%20conjecture.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e8/31/e831a001a1f865c5a08f5639e87fc148c7fd7d99.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.disc.2012.12.018"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Sharp lower bounds of the least eigenvalue of planar graphs

Yuan Hong, Jin-Long Shu
<span title="">1999</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/wsx3rzhpingfvewcn5nwhfkq3e" style="color: black;">Linear Algebra and its Applications</a> </i> &nbsp;
Let G be a simple graph with n P 3 vertices and orientable genus g and non-orientable genus h. We de®ne the Euler characteristic vq of a graph G by vq maxf2 À 2gY 2 À hg.  ...  Let kq be the least eigenvalue of the adjacency matrix A of G.  ...  Acknowledgements The authors are grateful to the referees for many helpful suggestions which led to an improved version of this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0024-3795(99)00129-9">doi:10.1016/s0024-3795(99)00129-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/lsnm4oaiyvf57kt6umvjf6mx7q">fatcat:lsnm4oaiyvf57kt6umvjf6mx7q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20030519051800/http://math.skku.ac.kr:80/~sglee/papers/pdf/LAA/article20000121-13.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/aa/7c/aa7cdacf4fdb64abb759b101cf6bffb563554fec.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0024-3795(99)00129-9"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

An obstruction to embedding graphs in surfaces

Bojan Mohar
<span title="">1989</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/civgv5utqzhu7aj6voo6vc5vx4" style="color: black;">Discrete Mathematics</a> </i> &nbsp;
It is shown that the genus of an embedding of a graph can be determined by the rank of a certain matrix. Several applications to problems involving the genus of graphs are presented.  ...  (b) If S is non-orientable then the non-orientable genus is y(S) = rank A. % . . . ) ek, be the consecutive edges on the boundary of C.  ...  Ringel [9] proved that the maximum non-orientable genus of any connected graph G is equal to its Betti number B(G) = IE(G)I -IV(G)1 + 1.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(89)90170-2">doi:10.1016/0012-365x(89)90170-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7wpseo2xszbw3l7e42vixcz6qq">fatcat:7wpseo2xszbw3l7e42vixcz6qq</a> </span>
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How to determine the maximum genus of a graph

Nguyen Huy Xuong
<span title="">1979</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/g6u5fful5vcr3a7gppc6y47el4" style="color: black;">Journal of combinatorial theory. Series B (Print)</a> </i> &nbsp;
Themaximum non-orientable genus, y&G), of a connected graph G is the largest non-orientable genus r(S) for non-orientable surfaces S in which G has a 2-cell embedding (recall that the Euler-characteristic  ...  with each connected graph G two non negative integers, y( (the genus of G) and yM(G) (the maximum genus of G>, such that G h Zcelluular embedding into an orientable surface S of genus y if and only if  ... 
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Asymptotic enumeration and limit laws for graphs of fixed genus [article]

Guillaume Chapuy, Eric Fusy, Omer Gimenez, Bojan Mohar, Marc Noy
<span title="2010-01-20">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S_g of genus g grows asymptotically like c^(g)n^5(g-1)/2-1γ^n n!  ...  It follows, in particular, that a random graph embeddable in S_g has a unique 2-connected component of linear size with high probability.  ...  The next step is to go to 3-connected graphs of non-orientable genus h via 3-connected maps on N h .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1001.3628v1">arXiv:1001.3628v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/yeer3tcy4nfm5izldjl3nzpngi">fatcat:yeer3tcy4nfm5izldjl3nzpngi</a> </span>
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Asymptotic enumeration and limit laws for graphs of fixed genus

Guillaume Chapuy, Éric Fusy, Omer Giménez, Bojan Mohar, Marc Noy
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/z77xaqun7bcxjkh75wb7iseaty" style="color: black;">Journal of combinatorial theory. Series A</a> </i> &nbsp;
It is shown that the number of labelled graphs with n vertices that can be embedded in the orientable surface S g of genus g grows asymptotically like c (g) n 5(g−1)/2−1 γ n n!  ...  It follows, in particular, that a random graph embeddable in S g has a unique 2-connected component of linear size with high probability.  ...  The next step is to go to 3-connected graphs of non-orientable genus h via 3-connected maps on N h .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jcta.2010.11.014">doi:10.1016/j.jcta.2010.11.014</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/sszfq3df65efbed3gnzjch4wfi">fatcat:sszfq3df65efbed3gnzjch4wfi</a> </span>
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