Labeling Schemes for Bounded Degree Graphs

We investigate adjacency labeling schemes for graphs of bounded degree $\Delta = O(1)$. In particular, we present an optimal (up to an additive constant) $\log n + O(1)$ adjacency labeling scheme for bounded degree trees. The latter scheme is derived from a labeling scheme for bounded degree outerplanar graphs. Our results complement a similar bound recently obtained for bounded depth trees [Fraigniaud and Korman, SODA 10], and may provide new insights for closing the long standing gap for adjacency in trees [Alstrup and Rauhe, FOCS 02]. We also provide improved labeling schemes for bounded degree planar graphs. Finally, we use combinatorial number systems and present an improved adjacency labeling schemes for graphs of bounded degree $\Delta$ with $(e+1)\sqrt{n} < \Delta \leq n/5$

Some Problems in Graph Coloring: Methods, Extensions and Results

The « Habilitation à 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 Lévêque, Amanda Montejano, Mickaël Montassier, Pascal Ochem, André Raspaud, Sagnik Sen and Éric 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, Frédéric Havet, Ross Kang, Daniel Král’, Colin McDiarmid, Michaël Rao, Jean-Sébastien Sereni and Stéphan Thomassé. 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 Lovász Local Lemma. The Lovász Local Lemma was introduced in the 70’s 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 Lovász 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 Lovász 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 Tardos’ 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 Lovász 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 Lovász 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 Nešetřil 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 ω to some elements of G. Using global information on the structure of G, we are able to compute the total sum of the charges ω(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 ω∗(G). We then get a contradiction by showing that ω(G) ̸= ω∗(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 “second generation” 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 ∆ + 1 colors for any 2-distance coloring since a vertex of degree ∆ together with its ∆ neighbors form a set of ∆ + 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 ∆, graphs with maximum degree ∆ and maximum average degree less that 3 − epsilon (for any epsilon > 0) admit a 2-distance coloring with ∆ + 1 colors. For maximum average degree less that 4 − epsilon, we prove that ∆ + C colors are enough (where C is a constant not depending on ∆). Finally, for maximum average degree at least 4, it is already known that C′∆ colors are enough. Therefore, thresholds of 3 − epsilon and 4 − epsilon are tight

Bandwidth, expansion, treewidth, separators, and universality for bounded degree graphs

We establish relations between the bandwidth and the treewidth of bounded degree graphs G, and relate these parameters to the size of a separator of G as well as the size of an expanding subgraph of G. Our results imply that if one of these parameters is sublinear in the number of vertices of G then so are all the others. This implies for example that graphs of fixed genus have sublinear bandwidth or, more generally, a corresponding result for graphs with any fixed forbidden minor. As a consequence we establish a simple criterion for universality for such classes of graphs and show for example that for each gamma>0 every n-vertex graph with minimum degree ((3/4)+gamma)n contains a copy of every bounded-degree planar graph on n vertices if n is sufficiently large

Sublinear Distance Labeling

A distance labeling scheme labels the $n$ nodes of a graph with binary strings such that, given the labels of any two nodes, one can determine the distance in the graph between the two nodes by looking only at the labels. A $D$-preserving distance labeling scheme only returns precise distances between pairs of nodes that are at distance at least $D$ from each other. In this paper we consider distance labeling schemes for the classical case of unweighted graphs with both directed and undirected edges. We present a $O(\frac{n}{D}\log^2 D)$ bit $D$-preserving distance labeling scheme, improving the previous bound by Bollob\'as et. al. [SIAM J. Discrete Math. 2005]. We also give an almost matching lower bound of $\Omega(\frac{n}{D})$. With our $D$-preserving distance labeling scheme as a building block, we additionally achieve the following results: 1. We present the first distance labeling scheme of size $o(n)$ for sparse graphs (and hence bounded degree graphs). This addresses an open problem by Gavoille et. al. [J. Algo. 2004], hereby separating the complexity from distance labeling in general graphs which require $\Omega(n)$ bits, Moon [Proc. of Glasgow Math. Association 1965]. 2. For approximate $r$-additive labeling schemes, that return distances within an additive error of $r$ we show a scheme of size $O\left ( \frac{n}{r} \cdot\frac{\operatorname{polylog} (r\log n)}{\log n} \right )$ for $r \ge 2$. This improves on the current best bound of $O\left(\frac{n}{r}\right)$ by Alstrup et. al. [SODA 2016] for sub-polynomial $r$, and is a generalization of a result by Gawrychowski et al. [arXiv preprint 2015] who showed this for $r=2$.Comment: A preliminary version of this paper appeared at ESA'1

Near-Optimal Induced Universal Graphs for Bounded Degree Graphs

A graph $U$ is an induced universal graph for a family $F$ of graphs if every graph in $F$ is a vertex-induced subgraph of $U$. For the family of all undirected graphs on $n$ vertices Alstrup, Kaplan, Thorup, and Zwick [STOC 2015] give an induced universal graph with $O\!\left(2^{n/2}\right)$ vertices, matching a lower bound by Moon [Proc. Glasgow Math. Assoc. 1965]. Let $k= \lceil D/2 \rceil$. Improving asymptotically on previous results by Butler [Graphs and Combinatorics 2009] and Esperet, Arnaud and Ochem [IPL 2008], we give an induced universal graph with $O\!\left(\frac{k2^k}{k!}n^k \right)$ vertices for the family of graphs with $n$ vertices of maximum degree $D$. For constant $D$, Butler gives a lower bound of $\Omega\!\left(n^{D/2}\right)$. For an odd constant $D\geq 3$, Esperet et al. and Alon and Capalbo [SODA 2008] give a graph with $O\!\left(n^{k-\frac{1}{D}}\right)$ vertices. Using their techniques for any (including constant) even values of $D$ gives asymptotically worse bounds than we present. For large $D$, i.e. when $D = \Omega\left(\log^3 n\right)$, the previous best upper bound was ${n\choose\lceil D/2\rceil} n^{O(1)}$ due to Adjiashvili and Rotbart [ICALP 2014]. We give upper and lower bounds showing that the size is ${\lfloor n/2\rfloor\choose\lfloor D/2 \rfloor}2^{\pm\tilde{O}\left(\sqrt{D}\right)}$. Hence the optimal size is $2^{\tilde{O}(D)}$ and our construction is within a factor of $2^{\tilde{O}\left(\sqrt{D}\right)}$ from this. The previous results were larger by at least a factor of $2^{\Omega(D)}$. As a part of the above, proving a conjecture by Esperet et al., we construct an induced universal graph with $2n-1$ vertices for the family of graphs with max degree $2$. In addition, we give results for acyclic graphs with max degree $2$ and cycle graphs. Our results imply the first labeling schemes that for any $D$ are at most $o(n)$ bits from optimal

On the spectral dimension of causal triangulations

We introduce an ensemble of infinite causal triangulations, called the uniform infinite causal triangulation, and show that it is equivalent to an ensemble of infinite trees, the uniform infinite planar tree. It is proved that in both cases the Hausdorff dimension almost surely equals 2. The infinite causal triangulations are shown to be almost surely recurrent or, equivalently, their spectral dimension is almost surely less than or equal to 2. We also establish that for certain reduced versions of the infinite causal triangulations the spectral dimension equals 2 both for the ensemble average and almost surely. The triangulation ensemble we consider is equivalent to the causal dynamical triangulation model of two-dimensional quantum gravity and therefore our results apply to that model.Comment: 22 pages, 6 figures; typos fixed, one extra figure, references update

Drawings of Planar Graphs with Few Slopes and Segments

We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected plane graph on $n$ vertices has a plane drawing with at most ${5/2}n$ segments and at most $2n$ slopes. We prove that every cubic 3-connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of non-planar graphs with few slopes are also considered.Comment: This paper is submitted to a journal. A preliminary version appeared as "Really Straight Graph Drawings" in the Graph Drawing 2004 conference. See http://arxiv.org/math/0606446 for a companion pape

Localization game on geometric and planar graphs

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph $G$ we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the \emph{metric dimension} of a graph. We provide upper bounds on the related graph invariant $\zeta (G)$, defined as the least number of cops needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth $2$ and unbounded $\zeta (G)$. On a positive side, we prove that $\zeta (G)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $\zeta (G)$ is NP-hard in graphs with diameter at most $2$. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane