Three years of graphs and music : some results in graph theory and its applications Nathann Cohen To cite this version: Nathann Cohen. Three years of graphs and music : some results in graph theory and its applications. Discrete Mathematics [cs.DM]. Université Nice Sophia Antipolis, 2011. English. tel-00645151 HAL Id: tel-00645151 https://tel.archives-ouvertes.fr/tel-00645151 Submitted on 26 Nov 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. UNIVERSITE´ DE NICE-SOPHIA ANTIPOLIS - UFR SCIENCES ECOLE´ DOCTORALE STIC SCIENCES ET TECHNOLOGIES DE L’INFORMATION ET DE LA COMMUNICATION TH ESE` pour obtenir le titre de Docteur en Sciences de l’Universite´ de Nice - Sophia Antipolis Mention : INFORMATIQUE Present´ ee´ et soutenue par Nathann COHEN Three years of graphs and music Some results in graph theory and its applications These` dirigee´ par Fred´ eric´ HAVET prepar´ ee´ dans le Projet MASCOTTE, I3S (CNRS/UNS)-INRIA soutenue le 20 octobre 2011 Rapporteurs : Jørgen BANG-JENSEN - Professor University of Southern Denmark, Odense Daniel KRAL´ ’ - Associate Professor, Charles University, Prague Andras´ SEBO˝ - Directeur de Recherches, G-SCOP Laboratory, Grenoble Directeur : Fred´ eric´ HAVET - Charge´ de Recherches, Mascotte, Sophia-Antipolis Examinateurs : Stephane´ BESSY - Maitre de Conferences,´ LIRMM, Montpellier Victor CHEPOI - Professeur, Universite´ de la Mediterranee,´ Luminy Ioan TODINCA - Professeur, LIFO, Universite´ d’Orleans´ Invite´ : Stephane´ PERENNES - Directeur de Recherches, Mascotte, Sophia-Antipolis 2 Three years of graphs and music Some results in graph theory and its applications Nathann Cohen 20 october 2011 2 Contents A few words 6 Pretty conjectures . 9 0 Coloring-oriented introduction to Graph Theory 11 0.1 Several graphs we will need . 12 0.2 Graph Coloring . 13 0.2.1 χ(G) seen from other parameters . 15 Maximum degree ∆(G) ............................ 16 Maximum clique ω(G) ............................. 17 Girth g(G) ................................... 18 Fractional Chromatic number χf (G) ..................... 19 0.3 Choosability . 20 0.3.1 Definition . 20 Bipartite graphs . 21 Certificates . 22 0.3.2 Nullstellensatz - Alon Tarsi - Bipartite graphs . 22 Choosability through the Nullstellensatz . 23 Bipartite Graphs . 24 A game version of choosability . 24 0.4 Edge coloring . 25 Vizing’s theorem . 26 Fractional Chromatic Index . 26 Obstructions - Overfull subgraphs . 27 Kempe chains . 28 Vizing’s theorem – Proof (Diestel’s Algorithm) . 28 List coloring conjecture . 29 Galvin’s proof for bipartite graphs – Choosability by kernels . 30 0.5 Planar graphs . 31 0.5.1 Maximum Average Degree . 32 Algorithmically . 33 Computing the maximum average degree by Linear Programming . 34 0.5.2 Girth . 35 0.5.3 Discharging . 35 3 4 CONTENTS 1 Acyclicity and colorings of planar graphs 37 1.1 Proper coloring . 37 1.2 Acyclicity . 39 Acyclic vertex coloring . 40 Acyclic edge coloring . 40 The 16∆ bound and the Local Lemma . 41 Actual computation . 43 1.2.1 Linear arboricity . 43 The intersection of three matroids . 44 1.2.2 Acyclic edge coloring of planar graphs . 46 1.2.3 Linear and 2-frugal choosability in graphs of small maximum average degree 47 2 Graph and Hypergraph decomposition 49 2.1 Arboricity in hypergraphs . 51 2.2 Induced decompositions . 55 2.2.1 Induced 2K2-decomposition of dense graphs . 56 2.2.2 Induced decomposition of dense graphs (second construction) . 57 2.2.3 Induced decompositions of dense graphs into small graphs H . 57 2.2.4 Enlarging our family of dense graphs . 59 2.2.5 Induced H-decompositions of dense graphs . 60 Using Frankl and Rodl’s¨ lemma . 61 3 Algorithms 63 3.1 Subgraph detection using color coding . 63 3.1.1 Color coding . 63 Derandomization – covering arrays . 64 3.1.2 FPT Complexity . 65 3.2 Reconfiguration problem . 66 Dependency digraph – second formulation . 67 Finding good reconfiguration orderings . 68 3.3 Good edge Labelling . 68 4 Wiener and Zagreb indices in chemical graphs 71 5 Applied theoretical research – Sage 75 5.1 Several new LP formulation of optimization problems . 78 Longest path . 80 Traveling Salesman Problem . 80 Steiner Tree . 80 Minor Testing . 80 Edge-disjoint spanning tree . 81 Linear arboricity . 81 Acyclic edge coloring . 81 CONTENTS 5 6 Appendices 83 Mixed Integer Linear Program formulations . 83 Acyclic edge-coloring of planar graphs . 91 Linear and 2-frugal choosability of graphs of small maximum average degree . 110 Planar graphs with maximum degree ∆ 9 are (∆ + 1)-edge-choosable – short proof . 125 ≥ The α-arboricity of complete uniform hypergraphs . 129 Algorithm for Finding k-Vertex Out-trees and its Application to k-Internal Out-branching Problem . 142 Tradeoffs in process strategy games with application in the WDM reconfiguration problem 161 Good Edge-Labelling of Graphs . 185 Graph Classes (Dis)satisfying the Zagreb Indices Inequality . 206 On Wiener index of graphs and their line graphs . 219 A note on Zagreb indices inequality for trees and unicyclic graphs . 230 Nomenclature 235 6 CONTENTS A few words This thesis is about three years happily spent playing with graphs, walking absentmindedly in cor- ridors or standing in front a whiteboard, while whistling or listening to music. It contains results obtained with several coauthors, as well as many of the elegant and more classical results I met reading books or papers, which to my sense gave a wider understanding of the problems presented. It has for background the laboratory of the MASCOTTE team set up by Jean-Claude Bermond and now headed by David Coudert, its offices whose doors open on whiteboards covered with neat and well-thought formulas – or hasty drawings depending on the character of the inhabitants – and its collection of open problems on which all of us have once tried our luck. I owe the time I spent on graph coloring to my advisor Fred´ eric´ Havet, on problems of hypergraphs to Jean-Claude Bermond, and the mixture of flows, graph decompositions and linear programming that helped me all along is definitely MASCOTTE’s own tune. I have met in Sophia-Antipolis and elsewhere quantity of people who never hesitated to sit down and think whenever a question was asked, for the sheer pleasure of doing so. I thank them all whole- heartedly, and in particular Stephane Perennes who, unbeknownst to him, probably taught me all I know of graphs today by mixing together in the same sentences the most remote notions, legitimately “confusing” one word with another, leaving to me to notice days later that “indeed, in this context replacing the natural notion with this apprently unrelated one was the best way to look at the problem at hand”. Because I regularly contributed to the math software Sage [145] during the last two years, I also found a place inside of a much wider family of mathematicians, sticking together for their interest in translating into algorithms the words of mathematical theory. It probably would have taken me much longer, if not for them, to notice how deeply mathematics is an experimental science, and I can but wonder at how much I would have missed if not for those nights spent implementing, fixing, testing and rewriting many of the most elementary algorithms graph theory has to offer, and that I wrongly assumed I had understood before explaining them to a computer. My fellow Sage developpers I want to thank for their energy, for their infinite enthusiasm, for the discussions at any time of day or night thanks the timezones they cover, and for constantly daring me to work on the whole spectrum of mathematics at the same time. Hence, I am largely indebted to my coauthors from here and abroad, and to all of the genuinely curious people I have had the pleasure to work with along the years. Let me also thank Jørgen Bang- Jensen, Daniel Kral` and Andras´ Sebo˝ who kindly accepted to read this manuscript as well as Stephane´ Bessy, Victor Chepoi, and Ioan Todinca who were in my jury. Oh. And because I never miss an occasion to advertise good books – especially when it is out of place – let me mention many names that were always around during those three years of graphs and music : Alessandro Barricco, Carlos Castaneda, Michel Foucault, Allen Ginsberg, Hermann 7 8 CONTENTS Hesse, Jack Kerouac, Milan Kundera, Robert Pirsig, Henry-David Thoreau (Walden), Oscar Wilde, and Stefan Zweig. Just yesterday I finished “The Island” from Aldous Huxley, which was a fantasic read too. I can find no words for my mother, who by now is aware that the largest part of everything I do she taught me herself. Nathann CONTENTS 9 Pretty conjectures On this page are presented four conjectures which, along with many unsolved others represent the largest part of the time I spent working during those three years of Ph.D. Though these do not appear in the following pages, open questions are in many respects much more interesting that known results. As they taught me much and are, after all, probably the best of what I learned during all these years, it would have been terribly ungrateful not to give them some space in this document. Chvatal’s´ conjecture This conjecture first appeared in 1974 in an article from Vasekˇ Chvatal´ [48]. Given a hereditary Set System, i.e. a family of sets 2X such that S any subset of F ⊆ ∀ ∈ F S is also an element of , one obtains an intersecting family (i.e.