123 research outputs found
Some Problems in Graph Coloring: Methods, Extensions and Results
The « Habilitation aÌ Diriger des Recherches » is the occasion to look back on my research work since the end of my PhD thesis in 2006. I will not present all my results in this manuscript but a selection of them: this will be an overview of eleven papers which have been published in international journals or are submitted and which are included in annexes. These papers have been done with different coauthors: Marthe Bonamy, Daniel Gonçalves, Benjamin LeÌveÌque, Amanda Montejano, MickaeÌl Montassier, Pascal Ochem, AndreÌ Raspaud, Sagnik Sen and EÌric Sopena. I would like to thanks them without whom this work would never have been possible. I also take this opportunity to thank all my other co-authors: Luigi Addario-Berry, François Dross, Louis Esperet, FreÌdeÌric Havet, Ross Kang, Daniel KraÌlâ, Colin McDiarmid, MichaeÌl Rao, Jean-SeÌbastien Sereni and SteÌphan ThomasseÌ. Working with you is always a pleasure !Since the beginning of my PhD, I have been interested in various fields of graph theory, but the main topic that I work on is the graph coloring. In particular, I have studied problems such as the oriented coloring, the acyclic coloring, the signed coloring, the square coloring, . . . It is then natural that this manuscript gathers results on graph coloring. It is divided into three chapters. Each chapter is dedicated to a method of proof that I have been led to use for my research works and that has given results described in this manuscript. We will present each method, some extensions and the related results. The lemmas, theorems, and others which I took part are shaded in this manuscript.# The entropy compression method.In the first chapter, we present a recent tool dubbed the entropy compression method which is based on the LovaÌsz Local Lemma. The LovaÌsz Local Lemma was introduced in the 70âs to prove results on 3-chromatic hypergraphs [EL75]. It is a remarkably powerful probabilistic method to prove the existence of combinatorial objects satisfying a set of constraints expressed as a set of bad events which must not occur. However, one of the weakness of the LovaÌsz Local Lemma is that it does not indicate how to efficiently avoid the bad events in practice.A recent breakthrough by Moser and Tardos [MT10] provides algorithmic version of the LovaÌsz Local Lemma in quite general circumstances. To do so, they used a new species of monotonicity argument dubbed the entropy compression method. This Moser and Tardosâ result was really inspiring and Grytczuk, Kozik and Micek [GKM13] adapted the technique for a problem on combi- natorics on words. This nice adaptation seems to be applicable to coloring problems, but not only, whenever the LovaÌsz Local Lemma is, with the benefits of providing better bounds. For example, the entropy compression method has been used to get bounds on non-repetitive coloring [DJKW14] that improve previous results using the LovaÌsz Local Lemma and on acyclic-edge coloring [EP13].In this context, we developed a general framework that can be applied to most of coloring problems. We then applied this framework and we get the best known bounds, up to now, for the acyclic chromatic number of graphs with bounded degree, non-repetitive chromatic number of graphs with bounded degree, facial Thue chromatic index of planar graphs, ... We also applied the entropy compression method to problems on combinatorics on words: we recently solved an old conjecture on pattern avoidance.# Graph homomorphisms and graph coloringsIn this chapter, we present some notions of graph colorings from the point of view of graph homomorphisms. It is well-known that a proper k-coloring of a simple graph G corresponds to a homomorphism of G to Kk. Considering homomorphisms from a more general context, we get a natural extension of the classical notion of coloring. We present in this chapter the notion of homomorphism of (n,m)-colored mixed graphs (graphs with arcs of n different types and edges of m different types) and the related notions of coloring. This has been introduced by NesÌetrÌil and Raspaud [NR00] in 2000 as a generalization of the classical notion of homomorphism. We then present two special cases, namely homomorphisms of (1, 0)-colored mixed graphs (which are known as oriented homomorphisms) and homomorphisms of (0,2)-colored mixed graphs (which are known as signed homomorphisms).While dealing with homomorphisms of graphs, one of the important tools is the notion of universal graphs: given a graph family F, a graph H is F-universal if each member of F admits a homomorphism to H. When H is F-universal, then the chromatic number of any member of F is upper-bounded by the number of vertices of H. We study some well-known families of universal graphs and we list their structural properties. Using these properties, we give some results on graph families such as bounded degree graphs, forests, partial k-trees, maximum average degree bounded graphs, planar graphs (with given girth), outerplanar graphs (with given girth), . . .Among others, we will present the Tromp construction which defines well known families of oriented and signed universal graphs. One of our major contributions is to study the properties of Tromp graphs and use them to get upper bounds for the oriented chromatic number and the signed chromatic number. In particular, up to now, we get the best upper bounds for the oriented chromatic number of planar graphs with girth 4 and 5: we get these bounds by showing that every graph of these two families admits an oriented homomorphism to some Tromp graph. We also get tight bounds for the signed chromatic number of several graph families, among which the family of partial 3-trees which admits a signed homomorphism to some Tromp graph.# Coloring the square of graphs with bounded maximum average degree using the discharging methodThe discharging method was introduced in the early 20th century, and is essentially known for being used by Appel, Haken and Kock [AH77, AHK77] in 1977 in order to prove the Four- Color-Theorem. More precisely, this technique is usually used to prove statements in structural graph theory, and it is commonly applied in the context of planar graphs and graphs with bounded maximum average degree.The principle is the following. Suppose that, given a set S of configurations, we want to prove that a graph G necessarily contains one of the configuration of S. We assign a charge Ï to some elements of G. Using global information on the structure of G, we are able to compute the total sum of the charges Ï(G). Then, assuming G does not contain any configuration from S, the discharging method redistributes the charges following some discharging rules (the discharging process ensures that no charge is lost and no charge is created). After the discharging process, we are able to compute the total sum of the new charges Ïâ(G). We then get a contradiction by showing that Ï(G) Ìž= Ïâ(G).Initially, the discharging method was used as a local discharging method. This means that the discharging rules was designed so that an element redistributes its charge in its neighborhood. However, in certain cases, the whole graph contains enough charge but this charge can be arbitrarily far away from the elements that are negative. In the last decade, the global discharging method has been designed. This notion of global discharging was introduced by Borodin, Ivanova and Kostochka [BIK07]. A discharging method is global when we consider arbitrarily large structures and make some charges travel arbitrarily far along those structures. In some sense, these techniques of global discharging can be viewed as the start of the âsecond generationâ of the discharging method, expanding its use to more difficult problems.The aim of this chapter is to present this method, in particular some progresses from the last decade, i.e. global discharging. To illustrate these progresses, we will consider the coloring of the square of graphs with bounded maximum average degree for which we obtained new results using the global discharging method. Coloring the square of a graph G consists to color its vertices so that two vertices at distance at most 2 get distinct colors (i.e. two adjacent vertices get distinct colors and two vertices sharing a common neighbor get distinct colors). This clearly corresponds to a proper coloring of the square of G. This coloring is called a 2-distance coloring. It is clear that we need at least â + 1 colors for any 2-distance coloring since a vertex of degree â together with its â neighbors form a set of â + 1 vertices which must get distinct colors. We investigate this coloring notion for graphs with bounded maximum average degree and we characterize two thresholds. We prove that, for sufficiently large â, graphs with maximum degree â and maximum average degree less that 3 â epsilon (for any epsilon > 0) admit a 2-distance coloring with â + 1 colors. For maximum average degree less that 4 â epsilon, we prove that â + C colors are enough (where C is a constant not depending on â). Finally, for maximum average degree at least 4, it is already known that CâČâ colors are enough. Therefore, thresholds of 3 â epsilon and 4 â epsilon are tight
Commutative Algorithms Approximate the LLL-distribution
Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local
Lemma (LLL), a series of works have exploited a key ingredient of the original
analysis, the witness tree lemma, in order to: derive deterministic, parallel
and distributed algorithms for the LLL, to estimate the entropy of the output
distribution, to partially avoid bad events, to deal with super-polynomially
many bad events, and even to devise new algorithmic frameworks. Meanwhile, a
parallel line of work, has established tools for analyzing stochastic local
search algorithms motivated by the LLL that do not fall within the Moser-Tardos
framework. Unfortunately, the aforementioned results do not transfer to these
more general settings. Mainly, this is because the witness tree lemma,
provably, no longer holds. Here we prove that for commutative algorithms, a
class recently introduced by Kolmogorov and which captures the vast majority of
LLL applications, the witness tree lemma does hold. Armed with this fact, we
extend the main result of Haeupler, Saha, and Srinivasan to commutative
algorithms, establishing that the output of such algorithms well-approximates
the LLL-distribution, i.e., the distribution obtained by conditioning on all
bad events being avoided, and give several new applications. For example, we
show that the recent algorithm of Molloy for list coloring number of sparse,
triangle-free graphs can output exponential many list colorings of the input
graph
Another approach to non-repetitive colorings of graphs of bounded degree
We propose a new proof technique that aims to be applied to the same problems
as the Lov\'asz Local Lemma or the entropy-compression method. We present this
approach in the context of non-repetitive colorings and we use it to improve
upper-bounds relating different non-repetitive numbers to the maximal degree of
a graph. It seems that there should be other interesting applications to the
presented approach.
In terms of upper-bound our approach seems to be as strong as
entropy-compression, but the proofs are more elementary and shorter. The
application we provide in this paper are upper bounds for graphs of maximal
degree at most : a minor improvement on the upper-bound of the
non-repetitive number, a upper-bound on the weak total
non-repetitive number and a
upper-bound on the total non-repetitive number of graphs. This last result
implies the same upper-bound for the non-repetitive index of graphs, which
improves the best known bound
Distributed Algorithms, the Lov\'{a}sz Local Lemma, and Descriptive Combinatorics
In this paper we consider coloring problems on graphs and other combinatorial
structures on standard Borel spaces. Our goal is to obtain sufficient
conditions under which such colorings can be made well-behaved in the sense of
topology or measure. To this end, we show that such well-behaved colorings can
be produced using certain powerful techniques from finite combinatorics and
computer science. First, we prove that efficient distributed coloring
algorithms (on finite graphs) yield well-behaved colorings of Borel graphs of
bounded degree; roughly speaking, deterministic algorithms produce Borel
colorings, while randomized algorithms give measurable and Baire-measurable
colorings. Second, we establish measurable and Baire-measurable versions of the
Symmetric Lov\'{a}sz Local Lemma (under the assumption
, which is stronger than the standard
LLL assumption but still sufficient
for many applications). From these general results, we derive a number of
consequences in descriptive combinatorics and ergodic theory.Comment: 35 page
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