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Global Discharging Methods for Coloring Problems in Graphs Marthe Bonamy

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Global Discharging Methods for Coloring Problems in Graphs Marthe Bonamy Global discharging methods for coloring problems in graphs Marthe Bonamy To cite this version: Marthe Bonamy. Global discharging methods for coloring problems in graphs. Discrete Mathematics [cs.DM]. Université de Montpellier, 2015. English. tel-02083632 HAL Id: tel-02083632 https://tel.archives-ouvertes.fr/tel-02083632 Submitted on 29 Mar 2019 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. Délivré par l’Université Montpellier Préparée au sein de l’école doctorale I2S Et de l’unité de recherche LIRMM Spécialité: INFORMATIQUE Présentée par Marthe Bonamy Global discharging methods for coloring problems in graphs Soutenue le 9 Février 2015 devant le jury composé de Rapporteur Maria CHUDNOVSKY, Professeur, Columbia University (Absente excusée) Roland DUCOURNAU, Professeur, UM2, LIRMM Examinateur Cyril GAVOILLE, Professeur, Université de Bordeaux, LaBRI Examinateur Frédéric HAVET, Directeur de Recherche, CNRS, I3S Rapporteur Dan KRÁL’, Professeur, Warwick University Rapporteur Benjamin LÉVÊQUE, Chargé de Recherche, CNRS, LIRMM Directeur Alexandre PINLOU, Maître de Conférences, UM3, LIRMM Directeur Présidé par Frédéric HAVET. Contents Quelques mots de français 5 Introduction 10 A few words on graph coloring . 11 Global picture . 13 1 Preliminaries 15 1.1 Basic introduction to graphs and coloring . 15 1.2 Definitions and notation . 24 1.3 What is a discharging argument? . 26 2 Illustrated Discharging Methods 33 2.1 Local arguments . 34 2.2 Global arguments . 43 2.3 Some remarks on the discharging method . 48 3 Edge Coloring 49 3.1 An overview of edge coloring . 49 3.2 Every planar graph G with ∆(G) ≥ 7 and no triangle adjacent to a C4 is edge ∆(G)-choosable . 53 3.3 Every planar graph G with ∆(G) ≥ 8 is edge (∆(G) + 1)-choosable . 64 4 Square coloring 95 4.1 An overview of square coloring . 95 4.2 A Brooks-like theorem on powers of graphs . 102 4.3 The threshold for (∆ + 1)-coloring squares of sparse graphs is 3 . 106 4.4 The threshold for (∆ + O(1))-coloring squares of sparse graphs is 4 . 113 4.5 (∆ + 1)-coloring squares of very sparse graphs . 115 4.6 (∆ + 2)-coloring squares of sparse graphs . 121 5 Conclusion 141 Publications . 142 Bibliography 145 3 Remerciements Merci, bien évidemment, à mes encadrants, Alex et Benjamin, pour tout ce qu’ils ont cherché à me transmettre, pour m’avoir permis d’aller à la rencontre des autres chercheurs, pour m’avoir, en somme, laissé tant de liberté et accordé tant d’attention. Merci à ceux qui m’ont fait l’honneur d’être mes rapporteurs. Dan (Král’), you were the first to trust me with a review. It meant a lot to me that you agreed to review this thesis. Fred (Havet), ma muse auto-proclamée (qu’ajouter ?). Maria (Chudnovsky), for all your good influence in the field. Un grand merci aussi à mes examinateurs, Roland (Ducournau) et Cyril (Gavoille). Merci à AlGCo, qui est objectivement, de toutes les équipes où j’ai fait une thèse, la meilleure et de loin. Par qui commencer sinon Nico (alias Dr Bousquet), sans qui ma thèse aurait été très différente (ça va bientôt faire un an, mais je me surprends encore parfois à regretter l’époque bénie où tu étais à portée de voix1). Mickaël (Montassier), qui a toujours su me redonner le sourire – merci pour tout ça. Stéphane (Bessy), qui est naturellement cool. Boris (Albar), pour les noix de coco et le reste2. François (Dross), mon petit frère de thèse, parce que c’était pas facile d’abandonner mon statut de thésarde unique (mais ça en valait le coup). Pascal (Ochem), fidèle à lui-même. Kévin (Sol), pour toute son humanité. Daniel (Gonçalves), l’éternel blagueur avec un coeur en or. Julien (Baste), pour les bananes et la vie en orange fluo, Valentin (Gar- nero), avec qui je n’ai toujours pas parlé de FPT (un jour un jour), Christophe (Paul), qui a failli figurer au rang des encadrants – tu auras les orangettes que je t’ai promises ! Ignasi (Sau), pour ton invariable enthousiasme, Guillaume (Guégan), ô chevalier de l’Esperluette, Eunjung (Kim), for trying to save me from workaholism, Mathias (Weller), pour avoir étendu ma culture musicale, Kolja (Knauer), pour les dégusta- tions de chocolat, Guilherme (D. da Fonseca), Dimitrios (Thilikos), Émeric (Gioan), Matthew (Francis), Philippe (Janssen). J’espère que vous continuerez longtemps à nar- guer tout le monde avec les copieux repas de Noël et autres manifestations. Merci à tous mes camarades thésards, Guillaume (Tisserant), pour nos discussions muettes, et les pauses thé qui faisaient disparaître tous les problèmes, Swan (Rocher), pour ta vision de la vie, Rémi (Watrigant), Guillerme (Duvillié), Léa (Guizol), Namrata (Pa- tel), Fabien (Hervouet), Mélanie (König), Sabrina (Ouazzani), Clément (Sipitier) - oui, Clément, je sais. Merci, plus généralement, aux équipes Maore, GraphiK et Texte. Merci à tous mes collègues enseignants (dont Sylvain Daudé, qui ne cesse de m’inspirer !), en particulier à ceux qui ont pris soin de mes élèves quand j’étais trop loin. Merci, aussi et surtout, à mes étudiants, que j’ai pour certains eu le plaisir de voir évoluer tout au long de ma thèse. Je ne pense pas que deux de mes groupes aient connu la même enseignante ; j’ai essayé de m’adapter tout au long de mon mon- itorat. Merci notamment à C. (qui se reconnaîtra), Valentin, Ulysse, Stella, Axel, Joce- lyn, Samuel, Elora, Sofien, Solène, et naturellement Emmanuel, pour toute la fraîcheur que vous avez apportée dans mon quotidien de thésarde. Merci à tous ceux qui m’ont aidée sur le plan administratif - je suis particulièrement reconnaissante envers Nico (Serrurier), Laurie (Lavernhe) et Caroline (Ycre). 1Depuis, on a quand même réussi à faire deux projets exclusivement par mail, alors je ne m’inquiète pas trop pour notre collaboration :) 2"Objection à la science ; le monde ne mérite pas d’être compris." 4 Merci aux chercheurs du monde entier qui m’ont fait me sentir une des leurs, à tous ceux qui font de notre communauté un groupe si vivant et accueillant. La thèse est aussi, en filigrane, une quête où on cherche son rôle dans le milieu de la recherche, avec ce que cela implique de doutes et de questions existentielles. Merci à tous ceux qui, consciemment ou inconsciemment, m’ont aidée dans cette quête. Merci à Stéphan Thomassé, entre tous, pour ta détermination à obtenir le meilleur de moi-même, to Ross Kang, who has provided me with consistently good advice for longer than any- one else (Beware the earless rabbit!), à Hervé Hocquard, avec qui j’ai toujours au- tant de plaisir à travailler, à Louis Esperet, grâce à qui j’ai bifurqué vers la théorie des graphes, to Łukasz Kowalik, who never ceases to impress me with his enthousi- asm and open-mindedness, to Tereza Klimošova, for the oranges we shared, à Aurélie Lagoutte, pour nos doutes parallèles, to my first graph theory group, in Durham more years ago than I care to count, Matthew Johnson, Ioannis Lignos, Viresh Patel and Daniel Paulusma, to the happy members of our bilateral european Proteus project, Riste Škrekovski, Borut Luczar, Matjaž Krnc, et enfin tous les (heureux !) hasards de rencontres qui ont donné lieu à des invitations ou collaborations ; Pierre Charbit, Luke Postle, Marcin Wrochna, Robert Lukotka, Arkadiusz Socała, Gwénaël Joret, Jakub Przybyło, Jean-Sébastien Sereni, Paul Seymour, Clément Charpentier, Marcin Kamin- sky, André Raspaud, Marin Bougeret, Samia Kerdjoudj, Dan Cranston, Michael Rao, Lucas Pastor, Andrew Treglown, Vincent Limouzy, Arnaud Pêcher, Michał Pilipczuk, Natasha Morrison. Je tiens également à saluer Petru Valicov, Arnaud Mary, Aline Par- reau, Pierre Aboulker, Simon Crevals, Mathieu Liedloff, Jan Volec, Jørgen Bang-Jensen, Jon Noel, Nicolas Trotignon, Julien Bensmail et Ararat Harutyunyan. Merci, même si ce n’est pas usuel, au système éducatif français qui m’a permis d’arriver jusque là quand rien ne m’y pré-destinait. Merci à la bourse au mérite at- tribuée par la région Haute-Normandie, puis au statut d’élève fonctionnaire-stagiaire normalien, pour la sérénité inattendue qu’ils m’ont offerte et qui m’a portée de mon bac jusqu’à l’entrée en thèse. Merci pour l’environnement enrichissant dans lequel j’ai pu évoluer. Merci en particulier à tous les enseignants qui ont cru en moi au fil de ma scolarité, et qui m’ont permis de rêver, progressivement, à ce que je vis aujourd’hui. Pour n’en citer qu’un, merci à Grégoire Taviot, qui a su m’amener à changer mon ap- proche des maths - j’en récolte encore aujourd’hui les fruits. Merci, enfin, à tous mes proches, pour leur soutien inconditionnel, nos discussions à bâtons rompus, les repas communs, les nuits blanches, les squats, la chaleur des sourires, et pour leurs inquiétudes, aussi. Merci à mes colocs, et aux colocs des autres, au groupe d’escargots qui tient depuis le collège, à ceux qui ont contribué au rituel de cookies sur un banc en prépa, aux petits chaperons rouges, au Raton-laveur, et à impega ; à Mao, Zabou, Adriane, Amaury, Hélène et Solène ; à mon caillou préféré, à Monsieur Robert, au plus beau chat du monde, à GRWGWR, et à tous les autres. Merci à ma famille ; à mes parents, mon frère, mes oncles et tantes, cousins et cousines, à ceux qui sont déjà partis et à ceux que l’on accueille, et surtout à mon compagnon, parce qu’on se comprend.
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