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Counterexamples to "A Conjecture on Induced Subgraphs of Cayley Graphs" [arXiv:2003.13166] [article]

Florian Lehner, Gabriel Verret
<span title="2020-05-21">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We disprove this conjecture by constructing various counterexamples, including an infinite family of Cayley graphs of unbounded valency which admit an induced subgraph of maximum valency 1 on a set of  ...  Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half its vertices has  ...  Acknowledgements Florian Lehner acknowledges the support of the Austrian Science Fund (FWF), grant J 3850-N32 and grant P 31889-N35. Gabriel Verret is grateful to the N.Z.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2004.01327v2">arXiv:2004.01327v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6fvqsqhdwjgi7odyeq44xyxepa">fatcat:6fvqsqhdwjgi7odyeq44xyxepa</a> </span>
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Counterexamples to "A conjecture on induced subgraphs of Cayley graphs"

Florian Lehner, Gabriel Verret
<span title="2020-03-25">2020</span> <i title="University of Primorska Press"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gvgtrk4iyvatbibvfwr3uifrji" style="color: black;">Ars Mathematica Contemporanea</a> </i> &nbsp;
We disprove this conjecture by constructing various counterexamples, including an infinite family of Cayley graphs of unbounded valency which admit an induced subgraph of maximum valency 1 on a set of  ...  Recently, Huang gave a very elegant proof of the Sensitivity Conjecture by proving that hypercube graphs have the following property: every induced subgraph on a set of more than half the vertices has  ...  As a consequence of the above remark, we can construct infinite families of d-regular Cayley graphs of order n admitting induced subgraphs of maximum degree 1 on 1+ε(d) 2 n vertices. 3.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26493/1855-3974.2301.63f">doi:10.26493/1855-3974.2301.63f</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nz23lxb2nfepngohpz3q356h3u">fatcat:nz23lxb2nfepngohpz3q356h3u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20201201130116/https://amc-journal.eu/index.php/amc/article/download/2301/1561" 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/f0/ac/f0ace0cc8fa68d3561dbbe822746986eed21ae25.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.26493/1855-3974.2301.63f"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Page 3814 of Mathematical Reviews Vol. , Issue 2002F [page]

<span title="">2002</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
In a 1967 paper, he gave constructions of several infinite families of such graphs.  ...  A map is regular if its automorphism group acts transitively on its flags. Let e = uv be an edge of a regular map M incident with a face f.  ... 
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Page 9579 of Mathematical Reviews Vol. , Issue 2004m [page]

<span title="">2004</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
We construct three infinite families of cubic edge-regular graphs.” 2004m:05132 05C25 05C20 05C70 Sato, Iwao [Sato, Iwao?] Isomorphisms of cyclic abelian covers of symmetric digraphs. II.  ...  If q is a prime power, the author constructs an infinite sequence of (q + 1)-regular Ramanujan graphs of bounded girth.  ... 
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Characterization and construction of Cayley graphs admitting regular Cayley maps

Robert Jajcay
<span title="">1996</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 the introduced characterization, we obtain some useful necessary conditions, as well as a constructive way of finding presentations of all Cayley graphs admitting regular Cayley maps. *  ...  The paper characterizes Cayley graphs that underlie Cayley maps of the highest possible level of symmetry, the regular Cayley maps, in terms of their automorphism groups.  ...  In the case of a balanced or antibalanced p this construction allows one to construct infinite families of regular Cayley maps.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0012-365x(95)00076-9">doi:10.1016/0012-365x(95)00076-9</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7wff4w43ifgfrhmoegbfcpjmy4">fatcat:7wff4w43ifgfrhmoegbfcpjmy4</a> </span>
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Construction of Directed Strongly Regular Graphs as Generalized Cayley Graphs [article]

Rongquan Feng, Liwei Zeng
<span title="2014-12-23">2014</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, an infinite family of directed strongly regular graphs is constructed, as generalized Cayley graphs of cyclic groups.  ...  Directed strongly regular graphs were introduced by Duval in 1998 as one of the possible generalization of classical strongly regular graphs to the directed case.  ...  An Infinite Family of DSRGs as Generalized Cayley Graphs of Cyclic Groups In this section, we will construct an infinite family of directed strongly regular graphs with parameters λ = µ = t − 1 as generalized  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1410.1161v2">arXiv:1410.1161v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/jesqb5fb2zfo7lf7mwzcpigrqu">fatcat:jesqb5fb2zfo7lf7mwzcpigrqu</a> </span>
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On symmetries of Cayley graphs and the graphs underlying regular maps

Marston Conder
<span title="">2009</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/h7qx4qsc2zf7hiupv27dke5fk4" style="color: black;">Journal of Algebra</a> </i> &nbsp;
By definition, Cayley graphs are vertex-transitive, and graphs underlying regular or orientably-regular maps (on surfaces) are arc-transitive.  ...  In particular, it is shown how to construct 3-valent Cayley graphs that are 5-arc-transitive (in answer to a question by Cai Heng Li), and Cayley graphs of valency 3 t + 1 that are 7-arc-transitive, for  ...  Further infinite families are described in [8] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jalgebra.2008.04.016">doi:10.1016/j.jalgebra.2008.04.016</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nirw5dsm7zchdiwa2bagt7lbtq">fatcat:nirw5dsm7zchdiwa2bagt7lbtq</a> </span>
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Infinitely many one-regular Cayley graphs on dihedral groups of any prescribed valency

Jin Ho Kwak, Young Soo Kwon, Ju-Mok Oh
<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;
A graph is one-regular if its automorphism group acts regularly on the arc set. In this paper, we construct a new infinite family of one-regular Cayley graphs of any prescribed valency.  ...  Also, with only finitely many possible exceptions, all of one-regular Cayley graphs on dihedral groups of any prescribed prime valency are constructed.  ...  Acknowledgments We would like to thank the referees of this paper for valuable comments.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.jctb.2007.09.005">doi:10.1016/j.jctb.2007.09.005</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/udrempgzy5e4tglboevfzfzsna">fatcat:udrempgzy5e4tglboevfzfzsna</a> </span>
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The topology of the minimal regular cover of the Archimedean tessellations [article]

Thierry Coulbois, Daniel Pellicer, Miguel Raggi, Camilo Ramírez, Ferrán Valdez
<span title="2012-10-04">2012</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this article we determine, for an infinite family of maps on the plane, the topology of the surface on which the minimal regular covering occurs. This infinite family includes all Archimedean maps.  ...  In this paper we show that the minimal regular cover of any map on a large family of maps on the Euclidean plane, including all Archimedean tessellations, lies on a topological surface obtained by glueing  ...  A regular cover M of the map M can be infinite (have infinitely many vertices, edges and faces) even when M is finite, and as a consequence Hartley's technique is hard to apply when attacking some concrete  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1210.1518v1">arXiv:1210.1518v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/m3hn7q2jsjdbrmplv4ll7qcgmi">fatcat:m3hn7q2jsjdbrmplv4ll7qcgmi</a> </span>
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Cayley Cages

Geoffrey Exoo, Robert Jajcay, Jozef Širáň
<span title="2012-10-04">2012</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cvausbtygzb5fdr6mvjzyvlh6e" style="color: black;">Journal of Algebraic Combinatorics</a> </i> &nbsp;
A (k, g)-Cayley cage is a k-regular Cayley graph of girth g and smallest possible order.  ...  We present an explicit construction of (k, g)-Cayley graphs for all parameters k ≥ 2 and g ≥ 3 and generalize this construction to show that many wellknown small k-regular graphs of girth g can be constructed  ...  The proof takes advantage of an infinite family of (infinite) Cayley maps constructed by Šiagiová and Watkins [13] and the fact that the automorphism groups of these maps are residually finite.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10801-012-0400-2">doi:10.1007/s10801-012-0400-2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/twau2n4hnvdrrjv2ljihjt3rve">fatcat:twau2n4hnvdrrjv2ljihjt3rve</a> </span>
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Page 4470 of Mathematical Reviews Vol. , Issue 95h [page]

<span title="">1995</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
For any prime power g, Morgenstern presents an explicit con- struction of infinite families A; of g + 1 regular Ramanujan graphs where the number of vertices of A; tends to infinity.  ...  (This is proved by constructing a remarkable map ¢ from 7(5; CS,,S) to the direct product of n —k copies of the segment A,,; although neither onto nor adjacency preserving, y induces a bijection between  ... 
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Edge connectivity in difference graphs and some new constructions of partial sum families

Alexander Araluze, Klavdija Kutnar, Luis Martínez, Dragan Marušič
<span title="">2011</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/54t3hgai4fhhthc74mj7z7tapu" style="color: black;">European journal of combinatorics (Print)</a> </i> &nbsp;
Finally, two infinite families of partial sum families that generate directed strongly regular graphs with new parameters are shown.  ...  In this paper, bounds for the edge connectivity of m-Cayley graphs are found, and also several structural conditions are given for a connected k-regular bi-abelian graph to have edge connectivity strictly  ...  A directed strongly regular graph will also be refereed to as a strongly regular digraph. Some infinite families of directed strongly regular graphs can be found in [6, 8, 9, 11, 12] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/j.ejc.2010.10.012">doi:10.1016/j.ejc.2010.10.012</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/x22s6bfca5e6xihg75wlkukfgi">fatcat:x22s6bfca5e6xihg75wlkukfgi</a> </span>
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Two families of graphs that are Cayley on nonisomorphic groups

Joy MORRİS, Josip SMOLCİC
<span title="2021-01-15">2021</span> <i title="Journal of Algebra Combinatorics Discrete Structures and Applications"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/3qhgdoxfkrcgrggfj5g24c3zpy" style="color: black;">Journal of Algebra Combinatorics Discrete Structures and Applications</a> </i> &nbsp;
In this paper, we construct two infinite families of graphs, each of which is Cayley on an abelian group and a nonabelian group.  ...  A number of authors have studied the question of when a graph can be represented as a Cayley graph on more than one nonisomorphic group.  ...  Acknowledgment: The authors thank the anonymous referees for helpful comments on this paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.13069/jacodesmath.867644">doi:10.13069/jacodesmath.867644</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qdmamp7t25c5vlhyk3lyeai7pi">fatcat:qdmamp7t25c5vlhyk3lyeai7pi</a> </span>
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Burnside's Problem, spanning trees and tilings

Brandon Seward
<span title="2014-01-23">2014</span> <i title="Mathematical Sciences Publishers"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nk6cbwjm3re3xcnnbdd2g7hpyy" style="color: black;">Geometry and Topology</a> </i> &nbsp;
In pursuit of these results we discover an interesting property of Cayley graphs: every finitely generated infinite group G has some Cayley graph having a regular spanning tree.  ...  This regular spanning tree can be chosen to have degree 2 (and hence be a bi-infinite Hamiltonian path) if and only if G has finitely many ends, and it can be chosen to have any degree greater than 2 if  ...  We will first construct a path on Γ which will be very similar to a bi-infinite Eulerian path.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.2140/gt.2014.18.179">doi:10.2140/gt.2014.18.179</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nkdspc2xfbgbrcadtmbvxcg47q">fatcat:nkdspc2xfbgbrcadtmbvxcg47q</a> </span>
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Kazhdan's Property (T) for Graphs [article]

Clara Brasseur, Ryan E. Grady, Stratos Prassidis
<span title="2006-07-14">2006</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In particular, we use our methods to construct infinite families of expanders as in the classical case.  ...  Through a combinatorial approach, we define an analogue of property (T) for regular graphs. We then prove the basic combinatorial and metric properties of Kazhdan groups in this context.  ...  Using this observation, a method of constructing families of expanders using Cayley graphs of infinite Kazhdan groups is described ( [11] , [14] ).  ... 
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