Aspects of Random Graphs Colourings, walkers and Hamiltonian cycles Tesi doctoral presentada al Programa de Doctorat de Matem`atica Aplicada de la Universitat Polit`ecnica de Catalunya per a optar al grau de Doctor en Matem`atiques per Xavier P´erez i Gim´enez sota la direcci´odel Doctor Josep D´ıaz i Cort Barcelona, juny de This thesis was presented at the Facultat de Matem`atiques i Estad´ıstica of the Universitat Polit`ecnica de Catalunya on June 22th, 2007. The dissertation committee consisted of: Marc Noy Serrano (Chair) — Departament de Matem`atica Aplicada ii, Universitat • Polit`ecnica de Catalunya. Mar´ıa Jos´eSerna Iglesias (Secretary) — Departament de Llenguatges i Sistemes • Inform`atics, Universitat Polit`ecnica de Catalunya. Lefteris M. Kirousis — Computer Engineering and Informatics Department, Univer- • sity of Patras. Nicholas C. Wormald — Department of Combinatorics and Optimization, University • of Waterloo. Colin McDiarmid — Departament of Statistics, University of Oxford. • Dedico aquest treball al Marrof´ı. Abstract This dissertation presents the author’s work in some problems involving different models of random graphs. First it contains a technical contribution towards solving the open problem of deciding whether with high probability a random 5-regular graph can be coloured with three colours. Next, the author proposes a model for the establishment and maintenance of communication between agents in a mobile ad-hoc network (manet), which is called the walkers model. We assume that the agents move through a fixed environment modelled by a motion graph, and are able to communicate only if they are at a distance of at most d. As the agents move randomly, we analyse how the connectivity between a set of w agents evolves over time, asymptotically for a large number N of vertices, when w also grows large. The particular topologies of the environment which are studied here are the cycle and the toroidal grid. For the latter, the results apply under any ℓ -normed distance, for 1 p . p ≤ ≤∞ Then, the dissertation follows with a continuous counterpart of the walkers model. Namely, it presents a model for manets based on random geometric graphs over the 2-dimensional unit torus, where each node moves under the random walk mobility model. More precisely, our model starts from a random geometric graph over the torus [0, 1)2, with n nodes and radius exactly at the connectivity threshold rc. Then each node chooses independently a random angle in [0, 2π) and moves for a number m of steps a fixed distance s > 0 in that direction. After these steps, each node again chooses a new angle and starts moving in that new direction, repeating the change of direction every m steps. We compute the expected number of steps for which the resulting graph stays connected or disconnected. In addition, for static random geometric graphs with radius at the connectivity threshold rc, we provide asymptotic expressions on the probability of existence of components according to their sizes. Finally, in the last part of this work, we show in a constructive way that, for any arbitrary ℓ -normed distance, 1 p , the property that a random geometric p ≤ ≤ ∞ graph under that distance contains a Hamiltonian cycle exhibits a sharp threshold at radius r = log n/(αpn), where αp is the area of the unit disk in the ℓp norm. p Resum Aquesta tesi presenta l’aportaci´ode l’autor en alguns problemes relacionats amb diferents models de grafs aleatoris. Primer cont´ela contribuci´ot`ecnica envers la soluci´odel problema obert de decidir si amb alta probabilitat un graf 5-regular aleatori pot ´esser acolorit amb tres colors. A continuaci´o, l’autor proposa un model per a l’establiment i manteniment de la comunicaci´oentre agents m`obils en una xarxa m`obil ad-hoc (manet), anomenat el model dels walkers. Suposem que els agents es mouen a trav´es d’un medi modelitzat per un graf motriu, i que s´on capa¸cos de comunicar-se entre ells si s´on a dist`ancia com a molt d. A mesura que els agents es belluguen a l’atzar, analitzem com evoluciona en el temps la connectivitat entre un conjunt de w agents, asimpt`oticament per a un gran nombre N de v`ertexs, quan w tamb´ecreix. Les topologies particulars del medi que estudiem aqu´ıs´on el cicle i la graella toro¨ıdal. En aquesta darrera, els resultats fan refer`encia a qualsevol dist`ancia normada ℓ , amb 1 p . Seguidament, la tesi continua amb una variant p ≤ ≤ ∞ cont´ınua del models dels walkers. Concretament, es presenta un model per a manets basat en grafs aleatoris geom`etrics sobre el torus unitat 2-dimensional, on cada node es belluga segons el model de mobilitat random walk. M´es detalladament, el nostre model parteix d’un 2 graf aleatori geom`etric en el torus [0, 1) , amb n nodes i radi exactament en el llindar rc de la connectivitat. Aleshores, cada node escull a l’atzar i de manera independent un angle de [0, 2π) i es mou durant m passes una dist`ancia fixada s > 0 en aquella direcci´o. Despr´es d’aquestes passes, tots els nodes escullen de nou un altre angle i comencen a moure’s cap all`a, repetint el canvi de direcci´ocada m passes. Es calcula el nombre esperat de passes durant les quals el graf resultant es mant´econnex o inconnex. A m´es, per als grafs aleatoris geom`etrics est`atics en el llindar de la connectivitat rc, donem expressions asimpt`otiques de la probabilitat d’exist`encia de components segons les seves talles. Finalment, en la darrera part d’aquest treball, mostrem de manera constructiva que, per a qualsevol dist`ancia normada ℓ arbitr`aria, 1 p , la propietat que un graf aleatori geom`etric contingui un cicle p ≤ ≤ ∞ Hamiltoni`aexhibeix un llindar abrupte en radi r = log n/(αpn), on αp ´es l’`area del disc unitat en la norma ℓ . p p Resumen Esta tesis presenta la aportaci´on del autor en algunos problemas relacionados con distintos modelos de grafos aleatorios. Primero contiene la contribuci´on t´ecnica hacia la soluci´on del problema abierto de decidir si con alta probabilidad un grafo 5-regular aleatorio puede ser coloreado con tres colores. A continuaci´on, el autor propone un modelo para el es- tablecimiento y mantenimiento de la comunicaci´on entre agentes m´oviles en una red m´ovil ad-hoc (manet), llamado el modelo de los walkers. Supongamos que los agentes se mueven a trav´es de un medio modelizado por un grafo motriz, y que son capaces de comunicarse entre ellos si est´an a distancia como mucho d. A medida que los agentes se mueven al azar, analizamos c´omo evoluciona en el tiempo la conectividad entre un conjunto de w agentes, asint´oticamente para un n´umero grande N de v´ertices, cuando w tambi´en crece. Las topolog´ıas particulares del medio que estudiamos aqu´ı son el ciclo y la malla toroidal. En ´esta ´ultima, los resultados se refieren a cualquier distancia normada ℓ , con 1 p . p ≤ ≤∞ Seguidamente, la tesis contin´ua con una variante continua del modelo de los walkers. Conc- retamente, se presenta un modelo para manets basado en los grafos aleatorios geom´etricos sobre el toro unidad 2-dimensional, donde cada nodo se mueve seg´un el modelo de movilidad random walk. M´as en detalle, nuestro modelo parte de un grafo aleatorio geom´etrico en el 2 toro [0, 1) , con n nodos y radio exactamente en el umbral rc de la conectividad. Entonces, cada nodo escoge al azar y de manera independiente un ´angulo de [0, 2π) y se mueve durante m Z pasos una distancia fijada s> 0 en aquella direcci´on. Despu´es de esos pasos, todos ∈ los nodos escogen de nuevo otro ´angulo y empiezan a moverse hacia all´ı, repitiendo el cambio de direcci´on cada m pasos. Se calcula el n´umero esperado de pasos durante los cuales el grafo resultante se mantiene conectado o desconectado. Adem´as, para los grafos aleatorios geom´etricos est´aticos en el umbral de la conectividad rc, damos expresiones asint´oticas de la probabilidad de existencia de componentes seg´un sus tallas. Finalmente, en la ´ultima parte de este trabajo, mostramos de manera constructiva que, para cualquier distancia normada arbitraria ℓ , 1 p , la propiedad de que un grafo aleatorio geom´etrico contenga un p ≤ ≤ ∞ ciclo Hamiltoniano presenta un umbral abrupto en radio r = log n/(αpn), donde αp es el ´area del disco unidad en la norma ℓ . p p Acknowledgements This dissertation could not have been completed without the generous assistance and advice of many people. For this reason, I would like to express my appreciation and gratitude. In the first place I would like to thank my thesis advisor Josep D´ıaz for his infinite patience and continuous encouragement. He always believed in me (more than I do myself), and devoted a great amount of energy to transmit his enthusiasm for research to me. I can positively assert that he achieved the goal. During these years, I had the privilege to collaborate with Nicholas Wormald. I have benefited enormously from working with him and have gained much insight and perspective into many probabilistic issues. I am also indebted to him and to his wife Hania for their warm hospitality in Waterloo. I also had the opportunity to work very closely with Lefteris Kirousis. I am very grateful to him for the wonderful time we spent together. During those days of intensive work, I had indeed great fun besides ØÓ learning a lot; sometimes research can be as thrilling as a suspense movie × Ø ÖØ Ñººº During the last stages of my thesis, I had the chance to work with Dieter Mitsche.