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ICGT 2018 10Th International Colloquium on Graph Theory and Combinatorics ICGT 2018 10th International Colloquium on Graph Theory and combinatorics July 9-13, 2018 Lyon, France Abstract book Contents Committees 5 Invited talks 7 Carsten Thomassen: From finite to countable to uncountable . .9 Daniela Kühn: On a conjecture of Erdos on locally sparse Steiner triple systems . 10 Zdeněk Dvořák: Graph Minors without the Structure Theorem . 11 Maria Chudnovsky: Trees and the Erdös-Hajnal conjecture . 12 László Babai: From local information to global symmetry: The Graph Isomorphism Problem . 13 László Babai: Hidden irregularity vs. hidden symmetry: The emer- gence of the Johnson graphs . 13 Samuel Fiorini: Graphs and extended formulations . 14 Daniel Lokshtanov: The number of things in things . 15 Claire Mathieu: Hierarchical Clustering: Objective Functions and Algorithms . 16 Daniel Král: Analytic methods in graph theory . 17 Selected talks 18 Committees Organizing committee: Scientific committee: Christophe Crespelle Cristina Bazgan (Paris) Éric Duchêne Pierre Charbit (Paris) Brice Effantin Victor Chepoi (Marseille) Guillaume Bagan Guillaume Fertin (Nantes) Mohamed Haddad Frédéric Havet (Sophia Antipolis) Hamamache Kheddouci (co-chair) Olivier Hudry (Paris) Bodo Lass Dieter Kratsch (Metz) Arnaud Mary Zhentao Li (Paris) Aline Parreau Yannis Manoussakis (Orsay) Michael Rao Christophe Paul (Montpellier) Hamida Seba Arnaud Pêcher (Bordeaux) Hamza Si Kaddour Myriam Preissmann (Grenoble) Éric Tannier Jean-Sébastien Sereni (Strasbourg) Éric Thierry Ian Todinca (Orléans) Stéphan Thomassé Olivier Togni (Dijon) Nicolas Trotignon (co-chair) Annegret Wagler (Clermont-Ferrand) Bora Ucar Rémi Watrigant 46 14 131 Amphi7 Amphi8 Amphi9 Amphi11 Amphi10 26 39 87 FRIDAY END OF ICGT 2018 ICGT ENDOF 101 109 125 Rooms: C. Mathieu D. Král 31 24 48 7 113 127 135 114 49 86 63 89 88 27 70 122 76 78 92 75 91 93 94 67 THURSDAY 3 6 25 44 68 107 100 118 S. Fiorini D. Lokshtanov 45 74 66 99 90 57 136 119 19 97 62 64 41 123 112 130 10 43 85 128 WEDNESDAY 12 13 30 35 SEMINAR - L. BABAI SEMINARL. - CONFERENCEDINNER ICGT 2018 ICGT SOCIAL EVENT : VISIT OF LYON OF VISIT : EVENT SOCIAL L. Babai 11 16 22 23 20 84 6 28 29 50 55 120 5 34 83 51 124 TUESDAY 95 102 111 104 108 OPEN PROBLEMS SESSION PROBLEMS OPEN Z. Dvořák M. Chudnovsky 80 81 98 17 56 10th International Colloquium on Graph Theory and combinatorics, Lyon, July 9-13, 9-13, 2018 July Lyon, combinatorics, and Theory Graph on Colloquium International 10th tba 137 4 61 60 42 82 103 115 110 32 40 47 36 73 132 133 134 MONDAY (Domus building) (Domus 8 9 54 15 52 58 59 33 Welcoming reception « COCKTAIL DINATOIRE «COCKTAIL » WELCOME - REGISTRATIONWELCOME - C. Thomassen D. Kühn 21 69 71 77 96 129 105 116 9h 10h 12h 14h 15h 18h 19h 8h30 9h30 11h45 10h30 10h45 15h15 16h05 16h30 17h30 17h45 Invited talks Carsten Thomassen: From finite to countable to uncountable Technical University of Denmark When: Monday morning A classical result discovered independently by Edmonds, Nash-Williams and Tutte says that every finite 2k-edge-connected graph has k pairwise edge- disjoint spanning trees. As a consequence, every finite 4k-edge-connected graph has a k-arc-connected orientation. Nash-Williams proved that 4k can be replaced by 2k. We discuss the possible extension of these and other basic results on finite graphs to the countable case or the uncountable case, in par- ticular a decomposition result of François Laviolette that allows extensions from the countable to the uncountable case. We also discuss the conjecture made by Erdős and Hajnal in 1966 that every graph of uncountable chromatic number has a subgraph of infinite connectivity. Daniel Soukup has obtained counterexamples to the vertex-connectivity version. We present (at least a sketch of) a proof of the edge-connectivity version of the conjecture. 9 Daniela Kühn: On a conjecture of Erdos on locally sparse Steiner triple systems University of Birmingham When: Monday afternoon Given a set X of size n, a collection S of 3-subsets of X is a Steiner triple system of order n if every 2-subset of X is contained in exactly one of the triples of S (so a Steiner triple system of order n can also be viewed as a decomposition of the n-vertex complete graph into edge-disjoint triangles). A famous theorem of Kirkman says that there exists a Steiner triple system of order n if and only if n equals 1,3 mod 6. In 1976, Erdos conjectured that one can find so-called “sparse” Steiner triple systems. Roughly speaking, the aim is to have at most j-3 triples in S on every set of j points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural gener- alization of the random triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and Rodl as well as Ellis and Linial in a strong form. Moreover, we pose a conjecture which would generalize the Erdos conjecture to Steiner systems with arbitrary parameters and provide some evidence for this. In my talk I will also survey some related decomposition problems. (joint work with Stefan Glock, Allan Lo and Deryk Osthus) 10 Zdeněk Dvořák: Graph Minors without the Structure Theo- rem Charles University, Prague When: Tuesday morning The Excluded Minor Structure Theorem of Robertson and Seymour is a powerful tool in the theory of proper minor-closed classes. On the flip side, its applications tend to be quite technical and lack real-world relevance due to huge constants involved in the process. We review several recent results on proper minor-closed classes that avoid these issues. 11 Maria Chudnovsky: Trees and the Erdös-Hajnal conjecture Princeton University When: Tuesday afternoon What is the effect of excluding an induced subgraph on the global struc- ture of a graph? While there do not seem to be general structural conse- quences, a conjecture of Erdos and Hajnal states that graphs with forbidden induced subgraphs behave very differently from general graphs; more pre- cisely they contain much larger cliques or stable sets. This conjecture is still open. We consider a stronger (but closely related) property. We say that a graph G has the "strong Erdos-Hajnal property" (with parameter epsilon) if it has two disjoint sets of vertices A and B, each of which contains epsilon- proportion of the vertex set of G, and such that either there are no edges between A and B, or every vertex of A is adjacent to every vertex of B. It is not hard to see that if all H-free graphs have the strong EH-property (with some fixed epsilon), then both H and Hc are forests, and Liebenau and Pilipczuk conjectured that this is, in some sense, if and only if. Their conjecture states that for every forest H, every (H; Hc)-free graph has the strong Erdos Hajnal property. Recently in joint work with Alex Scott, Paul Seymour and Sophie Spirkl we proved their conjecture. In this talk we will discuss the history of the problem, and some of the ideas of the proof. 12 László Babai: From local information to global symmetry: The Graph Isomorphism Problem University of Chicago When: Wednesday morning Deciding whether two finite graphs are isomorphic has long been known as one of a small number of natural computational problems with unsettled complexity status within the P/NP theory. We build on a group theoretic “Divide and Conquer” framework intro- duced in a seminal 1980 paper by Eugene M. Luks. Recent progress on the problem centers on an interplay between local information and global symmetry. The talk will attempt to illuminate this interplay. László Babai: Hidden irregularity vs. hidden symmetry: The emergence of the Johnson graphs University of Chicago When: Wednesday afternoon We describe in some detail the “Split-or-Johnson” routine, one of the combinatorial ingredients of the recent Graph Isomorphism tester. The al- gorithm either finds hidden irregularity in a graph or finds a canonically embedded highly symmetrical object: a Johnson graph. The work centers on highly regular partitions of the directed complete graph called “coherent configurations.” 13 Samuel Fiorini: Graphs and extended formulations Université libre de Bruxelles When: Thursday morning In the area of extended formulations, the main goal is to represent com- plicated convex sets as the projection of simpler convex sets. This is already interesting for polytopes. In order to quantify how succinctly a given poly- tope can be expressed as a projection of another polytope, one defines the extension complexity of a polytope P as the minimum number of facets of a polytope Q that projects to P. Many interesting polytopes are based on graphs (like for instance the spanning tree polytope, the matching polytope, or the stable set polytope), and it is interesting to see how the extension com- plexity of such polytopes depends on the structure of the graph. In the first part of my talk, I will give a quick overview of the field of extended formula- tions as we know it today, as well as state a few selected open problems, with a special emphasis on polytopes defined from graphs. In the second part, I will discuss how graphs appear beyond the scenes. In particular, I will discuss how they were used to obtain lower bounds on extension complexity or, in recent joint work with Marco Macchia and Kanstantsin Pashkovich, parametrize a class of polytopes that generalize stable set polytopes of per- fect graphs. 14 Daniel Lokshtanov: The number of things in things University of Bergen When: Thursday afternoon How many maximal independent sets can a graph on n vertices possibly have? Moon and Moser showed in 1965 that no graph can have more than 3n=3 maximal independent sets, and that there exist graphs which have this many.
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