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Colouring sparse graphs

François Pirot 1, 2 
1 ORPAILLEUR - Knowledge representation, reasonning
Inria Nancy - Grand Est, LORIA - NLPKD - Department of Natural Language Processing & Knowledge Discovery
Abstract : 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.
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Submitted on : Thursday, January 30, 2020 - 4:35:18 PM
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François Pirot. Colouring sparse graphs. Computer Science [cs]. Université de Lorraine; Radboud universiteit Nijmegen, 2019. English. ⟨NNT : 2019LORR0153⟩. ⟨tel-02461448⟩



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