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This thesis presents and develops a theory of self-reduction. This process is used to map instances of combinatorial optimisation problems onto smaller, more easily solvable instances in such a way that a solution of the former can be readily re-constructed, without loss of information or quality, from a solution of the latter. Self-reduction rules are surveyed for the Graph Colouring Problem, the Maximum Clique Problem, the Steiner Problem in Graphs, the Bin Packing Problem and the Set Covering Problem. This thesis introduces the problem of determining the maximum sequence of self-reductions on a given structure, and shows how the theory of confluence can be adapted from term re-writing to solve this problem by identifying rule sets for which all maximal reduction sequences are equivalent. Such confluence results are given for a number of reduction rules on problems on discrete systems. In contrast, NP-hardness results are also presented for some reduction rules. A probabilistic analysis of self-reductions on graphs is performed, showing that the expected number of self-reductions on a graph tends to zero as the order of the graph tends to infinity. An empirical study is performed comparing the performance of self-reduction, graph decomposition and direct methods of solving the Graph Colouring and Set Covering Problems. The results show that self-reduction is a potentially valuable, but sometimes erratic, method of finding exact solutions to combinatorial problems.

This thesis presents and develops a theory of self-reduction. This process is used to map instances of combinatorial optimisation problems onto smaller, more easily solvable instances in such a way that a solution of the former can be readily re-constructed, without loss of information or quality, from a solution of the latter. Self-reduction rules are surveyed for the Graph Colouring Problem, the Maximum Clique Problem, the Steiner Problem in Graphs, the Bin Packing Problem and the Set Covering Problem. This thesis introduces the problem of determining the maximum sequence of self-reductions on a given structure, and shows how the theory of confluence can be adapted from term re-writing to solve this problem by identifying rule sets for which all maximal reduction sequences are equivalent. Such confluence results are given for a number of reduction rules on problems on discrete systems. In contrast, NP-hardness results are also presented for some reduction rules. A probabilistic analysis of self-reductions on graphs is performed, showing that the expected number of self-reductions on a graph tends to zero as the order of the graph tends to infinity. An empirical study is performed comparing the performance of self-reduction, graph decomposition and direct methods of solving the Graph Colouring and Set Covering Problems. The results show that self-reduction is a potentially valuable, but sometimes erratic, method of finding exact solutions to combinatorial problems.

The classical (proper) graph colouring problem asks for a colouring of the vertices of a graph with the minimum number of colours such that no two vertices with the same colour are adjacent. Equivalently the colouring is required to be such that the graph induced by the vertices coloured the same colour has the maximum degree equal to zero. The graph parameter associated with the minimum possible number of colours of a graph is called chromatic number of that graph. One generalization of this classical problem is to relax the requirement that the maximum degree of the graph induced by the vertices coloured the same colour be zero, and instead allow it to be some integer d. For d = 0, we are back at the classical proper colouring. For other values of d we say that the colouring has defect d. Another generalization of the classical graph colouring, is list colouring and its associated parameters: choosability and choice number. The main result of this thesis is to show that every graph G of Euler genus μ is ⌈2 + √(3μ + 3)⌉–choosable with defect 1 (equivalently, with clustering 2). Thus allowing any defect, even 1, reduces the choice number of surface embeddable graphs below the chromatic number of the surface. For example, the chromatic number of the family of toroidal graphs is known to be 7. The bound above implies that toroidal graphs are 5-choosable with defect 1. This strengthens the result of Cowen, Goddard and Jesurum (1997) who showed that toroidal graphs are 5-colourable with defect 1. In a graph embedded in a surface, two faces that share an edge are called adjacent. We improve the above bound for graphs that have embeddings without adjacent triangles. In particular, we show that every non-planar graph G that can be embedded in a surface of Euler genus μ without adjacent triangles, is ⌈(5+ √(24μ + 1)) /3⌉–choosable with defect 1. This result generalizes the result of Xu and Zhang (2007) to all the surfaces. They proved that toroidal graphs that have embeddings on the torus without two adjacent triangles are 4-choosable with defect 1.

Cette thèse a pour thème la coloration de diverses classes de graphes épars. Shearer montra en 1983 [She83] que le ratio d'indépendance des graphes sans triangle de degré maximal d est au moins (1-o(1))ln d/d, et 13 ans plus tard Johansson [Joh96] démontra que le nombre chromatique de ces graphes est au plus O(d/ln d) quand d tend vers l'infini. Ce dernier résultat fut récemment amélioré par Molloy [Mol19], qui montra que la borne (1+o(1))d/ln d est valide quand d tend vers l'infini.Tandis que le résultat de Molloy s'exprime à l'aide d'un paramètre global, le degré maximal du graphe, nous montrons qu'il est possible de l'étendre à la coloration locale. Il s'agit de la coloration par liste, où la taille de la liste associée à chaque sommet ne dépend que de son degré. Avec une méthode différente se basant sur les propriétés de la distribution hard-core sur les ensembles indépendants d'un graphe, nous obtenons un résultat similaire pour la coloration fractionnaire locale, avec des hypothèses plus faibles. Nous démontrons également un résultat concernant la coloration fractionnaire locale des graphes où chaque sommet est contenu dans un nombre borné de triangles, et une borne principalement optimale sur le taux d'occupation — la taille moyenne des ensembles indépendants — de ces graphes. Nous considérons également les graphes de maille 7, et prouvons des résultats similaires qui améliorent les bornes précédemment connues quand le degré maximal du graphe est au plus 10^7. Finalement, pour les graphes d-réguliers où d vaut 3, 4, ou 5, de maille g variant entre 6 et 12, nous démontrons de nouvelles bornes inférieures sur le ratio d'indépendance.Le Chapitre 2 est dédié à la coloration à distance t d'un graphe, qui généralise la notion de coloration forte des arêtes. Nous cherchons à étendre le théorème de Johansson à la coloration à distance t, par l'exclusion de certains cycles. Le résultat de Johansson s'obtient par exclusion des triangles, ou des cycles de taille k pour n'importe quelle valeur de k. Nous montrons que l'exclusion des cycles de taille 2k, pour n'importe quel k>t, a un effet similaire sur le nombre chromatique à distance t, et sur l'indice chromatique à distance t+1. En outre, quand t est impair, une conclusion similaire peut se faire pour le nombre chromatique à distance t par l'exclusion des cycles de d'une taille impaire fixée valant au moins 3t. Nous étudions l'optimalité de ces résultats à l'aide de constructions de nature combinatoire, algébrique, et probabiliste.Dans le Chapitre 3, nous nous intéressons à la densité bipartie induite des graphes sans triangle, un paramètre relaxant celui de la coloration fractionnaire. Motivés par une conjecture de Esperet, Kang, et Thomassé [EKT19], qui prétend que la densité bipartie induite de graphes sans triangle de degré moyen d est au moins de l'ordre de ln d, nous démontrons cette conjecture quand d est suffisamment grand en termes du nombre de sommets n, à savoir d est au moins de l'ordre de (n ln n)^(1/2). Ce résultat ne pourrait être amélioré que par une valeur de l'ordre de ln n, ce que nous montrons à l'aide d'une construction reposant sur le processus sans triangle. Nos travaux se ramènent à un problème intéressant, celui de déterminer le nombre chromatique fractionnaire maximal d'un graphe épars à n sommets. Nous prouvons des bornes supérieures non triviales pour les graphes sans triangle, et pour les graphes dont chaque sommet appartient à un nombre borné de triangles.Cette thèse est reliée aux nombres de Ramsey. À ce jour, le meilleur encadrement connu sur R(3,t) nous est donné par le résultat de Shearer, et par une analyse récente du processus sans triangle [BoKe13+,FGM13+], ce qui donne(1-o(1)) t²/(4 ln t) < R(3,t) < (1+o(1)) t²/ln t. (1)Beaucoup de nos résultats ne pourraient être améliorés à moins d'améliorer par la même occasion (1), ce qui constituerait une révolution dans la théorie de Ramsey quantitative. / This thesis focuses on generalisations of the colouring problem in various classes of sparse graphs.Triangle-free graphs of maximum degree d are known to have independence ratio at least (1-o(1))ln d/d by a result of Shearer [She83], and chromatic number at most O(d/ln d) by a result of Johansson [Joh96], as d grows to infinity. This was recently improved by Molloy, who showed that the chromatic number of triangle-free graphs of maximum degree d is at most (1+o(1))d/ln d as d grows to infinity.While Molloy's result is expressed with a global parameter, the maximum degree of the graph, we first show that it is possible to extend it to local colourings. Those are list colourings where the size of the list associated to a given vertex depends only on the degree of that vertex. With a different method relying on the properties of the hard-core distribution on the independent sets of a graph, we obtain a similar result for local fractional colourings, with weaker assumptions. We also provide an analogous result concerning local fractional colourings of graphs where each vertex is contained in a bounded number of triangles, and a sharp bound for the occupancy fraction — the average size of an independent set — of those graphs. In another direction, we also consider graphs of girth 7, and prove related results which improve on the previously known bounds when the maximum degree does not exceed 10^7. Finally, for d-regular graphs with d in the set {3,4,5}, of girth g varying between 6 and 12, we provide new lower bounds on the independence ratio.The second chapter is dedicated to distance colourings of graphs, a generalisation of strong edge-colourings. Extending the theme of the first chapter, we investigate minimal sparsity conditions in order to obtain Johansson-like results for distance colourings. While Johansson's result follows from the exclusion of triangles — or actually of cycles of any fixed length — we show that excluding cycles of length 2k, provided that k>t, has a similar effect for the distance-t chromatic number and the distance-(t+1) chromatic index. When t is odd, the same holds for the distance-t chromatic number by excluding cycles of fixed odd length at least 3t. We investigate the asymptotic sharpness of our results with constructions of combinatorial, algebraic, and probabilistic natures.In the third chapter, we are interested in the bipartite induced density of triangle-free graphs, a parameter which conceptually lies between the independence ratio and the fractional chromatic number. Motivated by a conjecture of Esperet, Kang, and Thomassé [EKT19], which states that the bipartite induced density of a triangle-free graph of average degree d should be at least of the order of ln d, we prove that the conjecture holds for when d is large enough in terms of the number of vertices n, namely d is at least of the order of (n ln n)^(1/2). Our result is shown to be sharp up to term of the order of ln n, with a construction relying on the triangle-free process. Our work on the bipartite induced density raises an interesting related problem, which aims at determining the maximum possible fractional chromatic number of sparse graph where the only known parameter is the number of vertices. We prove non trivial upper bounds for triangle-free graphs, and graphs where each vertex belongs to a bounded number of triangles.All the content of this thesis is a collection of specialisations of the off-diagonal Ramsey theory. To this date, the best-known bounds on the off-diagonal Ramsey number R(3,t) come from the aforementioned result of Shearer for the upper-bound, and a recent analysis of the triangle-free process [BoKe13+,FGM13+] for the lower bound, giving(1-o(1)) t²/(4 ln t) < R(3,t) < (1+o(1)) t²/ln t. (1)Many of our results are best possible barring an improvement of (1), which would be a breakthrough in off-diagonal Ramsey theory.

Graph colourings and combinatorial games are two very widely studied topics in discrete mathematics. This thesis addresses the computational complexity of a range of problems falling within one or both of these subjects. Much of the thesis is concerned with the computational complexity of problems related to the combinatorial game (Free-)Flood-It, in which players aim to make a coloured graph monochromatic ("flood" the graph) with the minimum possible number of flooding operations; such problems are known to be computationally hard in many cases. We begin by proving some general structural results about the behaviour of the game, including a powerful characterisation of the number of moves required to flood a graph in terms of the number of moves required to flood its spanning trees; these structural results are then applied to prove tractability results about a number of flood-filling problems. We also consider the computational complexity of flood-filling problems when the game is played on a rectangular grid of fixed height (focussing in particular on 3xn and 2xn grids), answering an open question of Clifford, Jalsenius, Montanaro and Sach. The final chapter concerns the parameterised complexity of list problems on graphs of bounded treewidth. We prove structural results determining the list edge chromatic number and list total chromatic number of graphs with bounded treewidth and large maximum degree, which are special cases of the List (Edge) Colouring Conjecture and Total Colouring Conjecture respectively. Using these results, we show that the problem of determining either of these quantities is fixed parameter tractable, parameterised by the treewidth of the input graph. Finally, we analyse a list version of the Hamilton Path problem, and prove it to be W[1]-hard when parameterised by the pathwidth of the input graph. These results answer two open questions of Fellows, Fomin, Lokshtanov, Rosamond, Saurabh, Szeider and Thomassen.

We study graph parameters arising from different types of colourings of random graphs, defined broadly as an assignment of colours to either the vertices or the edges of a graph. The chromatic number X(G) of a graph is the minimum number of colours required for a vertex colouring where no two adjacent vertices are coloured the same. Determining the chromatic number is one of the classic challenges in random graph theory. In Chapter 3, we give new upper and lower bounds for the chromatic number of the dense random graph G(n,p)) where p &isin; (0,1) is constant. These bounds are the first to match up to an additive term of order o(1) in the denominator, and in particular, they determine the average colour class size in an optimal colouring up to an additive term of order o(1). In Chapter 4, we study a related graph parameter called the equitable chromatic number. This is defined as the minimum number of colours needed for a vertex colouring where no two adjacent vertices are coloured the same and, additionally, all colour classes are as equal in size as possible. We prove one point concentration of the equitable chromatic number of the dense random graph G(n,m) with m = pn(n-1)/2, p &LT; 1-1/e² constant, on a subsequence of the integers. We also show that whp, the dense random graph G(n,p) allows an almost equitable colouring with a near optimal number of colours. We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, which is a path where no colour is repeated. The least number of colours where this is possible is called the rainbow connection number rc(G). Since its introduction in 2008 as a new way to quantify how well connected a given graph is, the rainbow connection number has attracted the attention of a great number of researchers. For any graph G, rc(G)&ge;diam(G), where diam(G) denotes the diameter. In Chapter 5, we will see that in the random graph G(n,p), rainbow connection number 2 is essentially equivalent to diameter 2. More specifically, we consider G ~ G(n,p) close to the diameter 2 threshold and show that whp rc(G) = diam(G) &isin; {2,3}. Furthermore, we show that in the random graph process, whp the hitting times of diameter 2 and of rainbow connection number 2 coincide. In Chapter 6, we investigate sharp thresholds for the property rc(G)&le;=r where r is a fixed integer. The results of Chapter 6 imply that for r=2, the properties rc(G)&le;=2 and diam(G)&le;=2 share the same sharp threshold. For r&ge;3, the situation seems quite different. We propose an alternative threshold and prove that this is an upper bound for the sharp threshold for rc(G)&le;=r where r&ge;3.

In this thesis, we will discuss the construction of algorithms for solving Constraint Satisfaction Problems (CSPs), and describe two new ways of approaching them. Both approaches are based on the idea that it is sometimes faster to solve a large number of restricted problems than a single, large, problem. One of the strong points of these methods is that the intuition behind them is fairly simple, which is a definite advantage over many techniques currently in use. The first method, the covering method, can be described as follows: We want to solve a CSP with n variables, each having a domain with d elements. We have access to an algorithm which solves problems where the domain has size e &lt; d, and we want to cover the original problem using a number of restricted instances, in such a way that the solution set is preserved. There are two ways of doing this, depending on the amount of work we are willing to invest; either we construct an explicit covering and end up with a deterministic algorithm for the problem, or we use a probabilistic reasoning and end up with a probabilistic algorithm. The second method, called the partitioning method, relaxes the demand on the underlying algorithm. Instead of having a single algorithm for solving problems with domain less than d, we allow an arbitrary number of them, each solving the problem for a different domain size. Thus by splitting, or partitioning, the domain of the large problem, we again solve a large number of smaller problems before arriving at a solution. Armed with these new techniques, we study a number of different problems; the decision problems (d, l)-CSP and k-Colourability, together with their counting counterparts, as well as the optimisation problems Max Ind CSP, Max Value CSP, Max CSP, and Max Hamming CSP. Among the results, we find a very fast, polynomial space algorithm for determining k-colourability of graphs.

The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-hereditary families, and some generalisations, from a colouring perspective. Our main results are: · Additive induced-hereditary families are uniquely factorisable into irreducible families. · If <i>P</i> and <i>Q</i> are additive induced-hereditary graph families, then (<i>P</i>,<i>Q</i>)-COLOURING is NP-hard, with the exception of GRAPH 2-COLOURING. Moreover, with the same exception, (<i>P</i>,<i>Q</i>)-COLOURING is NP-complete iff <i>P</i>- and <i>Q</i>-RECOGNITION are both in NP. This proves a 1997 conjecture of Kratochvíl and Schiermeyer. We also provide generalisations to somewhat larger families. Other results that we prove include: · a characterisation of the minimal forbidden subgraphs of a hereditary property in terms of its minimal forbidden induced-subgraphs, and <i>vice versa</i>; · extensions of Mihók's construction of uniquely colourable graphs, and Scheinerman's characterisations of compositivity, to disjoint compositive properties; · an induced-hereditary property has at least two factorisations into arbitrary irreducible properties, with an explicitly described set of exceptions; · if <i>G</i> is a generating set for <i>A</i> &#959; <i>B</i>, where <i>A</i> and <i>B</i> are indiscompositive, then we can extract generating sets for <i>A</i> and <i>B</i> using a <i>greedy algorithm</i>.

The study and recognition of graph families (or graph properties) is an essential part of combinatorics. Graph colouring is another fundamental concept of graph theory that can be looked at, in large part, as the recognition of a family of graphs that are colourable according to certain rules. In this thesis, we study additive induced-hereditary families, and some generalisations, from a colouring perspective. Our main results are: · Additive induced-hereditary families are uniquely factorisable into irreducible families. · If <i>P</i> and <i>Q</i> are additive induced-hereditary graph families, then (<i>P</i>,<i>Q</i>)-COLOURING is NP-hard, with the exception of GRAPH 2-COLOURING. Moreover, with the same exception, (<i>P</i>,<i>Q</i>)-COLOURING is NP-complete iff <i>P</i>- and <i>Q</i>-RECOGNITION are both in NP. This proves a 1997 conjecture of Kratochvíl and Schiermeyer. We also provide generalisations to somewhat larger families. Other results that we prove include: · a characterisation of the minimal forbidden subgraphs of a hereditary property in terms of its minimal forbidden induced-subgraphs, and <i>vice versa</i>; · extensions of Mihók's construction of uniquely colourable graphs, and Scheinerman's characterisations of compositivity, to disjoint compositive properties; · an induced-hereditary property has at least two factorisations into arbitrary irreducible properties, with an explicitly described set of exceptions; · if <i>G</i> is a generating set for <i>A</i> &#959; <i>B</i>, where <i>A</i> and <i>B</i> are indiscompositive, then we can extract generating sets for <i>A</i> and <i>B</i> using a <i>greedy algorithm</i>.