Schedule-at-a-Glance

12 PRIMA 2013 AbstractsStructural limits in logical and analytic contextJaroslav NešetřilCharles University in Prague, Czech Republicnesetril@iuuk.mff.cuni.czWe survey recent development of limits of graphs (andother structures) in the unifying context of model theory.Particularly, this leads to the analysis of limits of sparsegraphs (with unbounded degrees). This is joint work withPatrice Ossona de Mendez (EHESS, Paris).Unavoidable vertex-minors in large primegraphsSang-il OumKorea Advanced Institute of Science and Technology, Koreasangil@kaist.eduA graph is prime (with respect to the split decomposition)if its vertex set does not admit a partition (A, B)(called a split) with |A|, |B| ≥ 2 such that if a pair of verticesin A have neighbors in B, then they have the sameset of neighbors in B. We prove that for each t, every sufficientlylarge prime graph must contain a vertex-minorisomorphic to either a cycle of length t or a graph consistingof two vertices joined by t 3-edge paths and no otheredges. This is a joint work with O-joung Kwon (KAIST).The structure of a typical H-free graphBruce ReedMcGill University, Canadabreed@cs.mcgill.caWe discuss the structure of a typical graph which does notcontain H as an induced subgraph and prove a weakeningof the Erdos-Hajnal conjecture. The talk presents jointwork with F. Havet, R. Kang, P. Keevash, M. Loebl, C.McDiarmid, J. Noel, A. Scott, and A. Thomason.Hamiltonian cycles in prismsMoshe RosenfeldUniversity of Washington, Tacoma, USAmoishe@uw.eduThe prism over a graph G is the Cartesian product ofG with K 2 . My interest in prisms began in 1973 whenI tried to tackle Dave Barnette’s conjecture (1970) thatall simple 4-polytopes are Hamiltonian (still open). Subsequently,we merged the study of Hamiltonian cycles inprisms with other refinements of Hamiltonian cycles. Weobserved that if G has a Hamiltonian prism then G hasa spanning closed 2-walk but the opposite is not true,that is having a Hamiltonian prism is "closer" to beingHamiltonian than having a spanning, closed 2-walk. Thisobservation created many opportunities to study variousclassical problems on Hamiltonicity of graphs.In this talk I will discuss some results on prism Hamiltonicity,further refinements of Hamiltonicity and relatedopen problems.On the computation of the Radon number insome graph convexitiesJayme L SzwarcfiterFederal University of Rio de Janeiro, Braziljayme@nce.ufrj.brA graph convexity is a pair (G, C), where G is a finitegraph with vertex V (G) and C a family of subsets ofV (G) satisfying ∅, V (G) ∈ C and being closed under intersections.The sets C ∈ C are called convex sets. Themost common graph convexities are those whose convexsets are defined through special paths of the graph.Among them the most prominent are the geodesic convexity,where C is closed under taking shortest paths, themonophonic convexity, where C is closed under inducedsubgraphs and the P 3 convexity, whose convex sets areclosed under pairs of common neighbors. In special, thelatter is closely related to some well studied graph processes,as percolation and conversion processes. We examinesome common parameters of graph convexities, as thegeodetic number, convexity number, hull number, Hellynumber, Carathéodory number and Radon number. Inparticular, we describe complexity results related to thecomputation of the Radon number. These include hardnessresults, polynomial-time algorithms and bounds.A class of periodic continued fractions andfactorization in modular groupMikhail TyaglovShanghai Jiao Tong University, Chinatyaglov@sjtu.edu.cnIn this talk we introduce some properties of periodiccontinued fractions of the form−1 −1 −1b 1 + b 2 +···+ b} {{ n}n+−1 −1 −1b 1 + b 2 +···+ b · · · , (1)} {{ n +}nwhere b k are (positive) integers. Such continued fractionsare called periodic negative-regular continuedfraction. By Tietze’s theorem [1], the fraction (1)converges to an irrational number if |b k | 2 (exceptfor the case |b k | = 2 for all k).We have proved that without the requirement |b k | 2 the fraction (1) may converge to rational numbersor diverge. This is the difference between negativeregularcontinued fractions and classical regular continuedfractions. The last one always converge toirrational numbers.Several algorithms for construction of periods{b 1, . . . , b n} of periodic negative-regular continuedfractions converging to rational numbers are given.The periods of a given length can be obtained by Fermat’sinfinite descent method applied to some Diophantineequations. An explicit simple formula forthe minimal period for x is presented. A constructionusing the Calkin-Wilf tree and Stern’s diatomic seriesis described. Arbitrary primitive periods are in oneto-onecorrespondence with elements of the modulargroup Γ. Explicit formulas converting products ofthe standard generators S and ST in Γ into primitiveperiods are obtained. The periods of elliptic elementsof Γ are completely described. This description resultsin a parametric formula for primitive periods ofrational numbers. A periodic negative-regular continuedfraction diverges if and only if either its periodor its double or its triple represents the identity inΓ.[1] H. Tietze, "Uber Kriterien für Konvergenz undIrrationalität unendlicher Ketenbrüche”, Math. Ann.70 (1911).The graph isomorphism problem on geometricgraphs