dc.contributor.author Dalfó Simó, Cristina dc.contributor.author Huemer, Clemens dc.contributor.author Salas Piñon, Julián dc.contributor.other Universitat Politècnica de Catalunya. Departament de Matemàtiques dc.date.accessioned 2016-09-14T10:33:44Z dc.date.available 2016-09-14T10:33:44Z dc.date.issued 2016 dc.identifier.citation Dalfo, C., Huemer, C., Salas, J. The degree/diameter problem in maximal planar bipartite graphs. "Electronic journal of combinatorics", 2016, vol. 23, núm. 1, p. 1-23. dc.identifier.issn 1077-8926 dc.identifier.uri http://hdl.handle.net/2117/89907 dc.description.abstract The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases. dc.format.extent 23 p. dc.language.iso eng dc.subject Àrees temàtiques de la UPC::Matemàtiques i estadística::Matemàtica discreta::Teoria de grafs dc.subject.lcsh Graph theory dc.subject.other Degree/diameter Problem dc.subject.other Planar graphs dc.subject.other Bipartite graphs dc.title The degree/diameter problem in maximal planar bipartite graphs dc.type Article dc.subject.lemac Grafs, Teoria de dc.contributor.group Universitat Politècnica de Catalunya. COMBGRAPH - Combinatòria, Teoria de Grafs i Aplicacions dc.contributor.group Universitat Politècnica de Catalunya. DCCG - Grup de recerca en geometria computacional, combinatoria i discreta dc.description.peerreviewed Peer Reviewed dc.subject.ams Classificació AMS::05 Combinatorics::05C Graph theory dc.relation.publisherversion http://www.combinatorics.org/ojs/index.php/eljc/article/view/v23i1p60/pdf dc.rights.access Open Access drac.iddocument 18535773 dc.description.version Postprint (published version) dc.relation.projectid info:eu-repo/grantAgreement/MICINN/1PE/MTM2014-60127-P upcommons.citation.author Dalfo, C., Huemer, C., Salas, J. upcommons.citation.published true upcommons.citation.publicationName Electronic journal of combinatorics upcommons.citation.volume 23 upcommons.citation.number 1 upcommons.citation.startingPage 1 upcommons.citation.endingPage 23
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