Ramsey multiplicity and the Tur\'an coloring

Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed graph $H$. This conjecture was disproved independently by Sidorenko and Thomason. The first author later found quantitatively stronger counterexamples, using the Tur\'an coloring, in which one of the two colors spans a balanced complete multipartite graph. We prove that the Tur\'an coloring is extremal for an infinite family of graphs, and that it is the unique extremal coloring. This yields the first determination of the Ramsey multiplicity constant of a graph for which the Burr--Rosta conjecture fails. We also prove an analogous three-color result. In this case, our result is conditional on a certain natural conjecture on the behavior of two-color Ramsey numbers.Comment: 37 page

Tur\'an Colourings in Off-Diagonal Ramsey Multiplicity

The Ramsey multiplicity constant of a graph $H$ is the limit as $n$ tends to infinity of the minimum density of monochromatic labelled copies of $H$ in a colouring of the edges of $K_n$ with two colours. Fox and Wigderson recently identified a large family of graphs whose Ramsey multiplicity constants are attained by sequences of "Tur\'an colourings;" i.e. colourings in which one of the colour classes forms the edge set of a balanced complete multipartite graph. The graphs in their family come from taking a connected non-3-colourable graph with a critical edge and adding many pendant edges. We extend their result to an off-diagonal variant of the Ramsey multiplicity constant which involves minimizing a weighted sum of red copies of one graph and blue copies of another. We also apply the flag algebra method to investigate the minimum number of pendant edges required for Tur\'an colourings to become optimal when the underlying graphs are small cliques.Comment: 48 pages, 2 figure

Recent developments in graph Ramsey theory

Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress

THE ELECTRONIC JOURNAL OF COMBINATORICS (2014), DS1.14 References

and Computing 11. The results of 143 references depend on computer algorithms. The references are ordered alphabetically by the last name of the first author, and where multiple papers have the same first author they are ordered by the last name of the second author, etc. We preferred that all work by the same author be in consecutive positions. Unfortunately, this causes that some of the abbreviations are not in alphabetical order. For example, [BaRT] is earlier on the list than [BaLS]. We also wish to explain a possible confusion with respect to the order of parts and spelling of Chinese names. We put them without any abbreviations, often with the last name written first as is customary in original. Sometimes this is different from the citations in other sources. One can obtain all variations of writing any specific name by consulting the authors database of Mathematical Reviews a

Characterizing and Detecting Unrevealed Elements of Network Systems

This dissertation addresses the problem of discovering and characterizing unknown elements in network systems. Klir (1985) provides a general definition of a system as “... a set of some things and a relation among the things (p. 4). A system, where the `things\u27, i.e. nodes, are related through links is a network system (Klir, 1985). The nodes can represent a range of entities such as machines or people (Pearl, 2001; Wasserman & Faust, 1994). Likewise, links can represent abstract relationships such as causal influence or more visible ties such as roads (Pearl, 1988, pp. 50-51; Wasserman & Faust, 1994; Winston, 1994, p. 394). It is not uncommon to have incomplete knowledge of network systems due to either passive circumstances, e.g. limited resources to observe a network, active circumstances, e.g. intentional acts of concealment, or some combination of active and passive influences (McCormick & Owen, 2000, p. 175; National Research Council, 2005, pp. 7, 11). This research provides statistical and graph theoretic approaches for such situations, including those in which nodes are causally related (Geiger & Pearl, 1990, pp. 3, 10; Glymour, Scheines, Spirtes, & Kelly, 1987, pp. 75-86, 178183; Murphy, 1998; Verma & Pearl, 1991, pp. 257, 260, 264-265). A related aspect of this research is accuracy assessment. It is possible an analyst could fail to detect a network element, or be aware of network elements, but incorrectly conclude the associated network system structure (Borgatti, Carley, & Krackhardt, 2006). The possibilities require assessment of the accuracy of the observed and conjectured network systems, and this research provides a means to do so (Cavallo & Klir, 1979, p. 143; Kelly, 1957, p. 968)

Extremal and Structural Problems of Graphs

In this dissertation, we are interested in studying several parameters of graphs and understanding their extreme values. We begin in Chapter~$2$ with a question on edge colouring. When can a partial proper edge colouring of a graph of maximum degree $\Delta$ be extended to a proper colouring of the entire graph using an `optimal' set of colours? Albertson and Moore conjectured this is always possible provided no two precoloured edges are within distance $2$. The main result of Chapter~$2$ comes close to proving this conjecture. Moreover, in Chapter~$3$, we completely answer the previous question for the class of planar graphs. Next, in Chapter~$4$, we investigate some Ramsey theoretical problems. We determine exactly what minimum degree a graph $G$ must have to guarantee that, for any two-colouring of $E(G)$, we can partition $V(G)$ into two parts where each part induces a connected monochromatic subgraph. This completely resolves a conjecture of Bal and Debiasio. We also prove a `covering' version of this result. Finally, we study another variant of these problems which deals with coverings of a graph by monochromatic components of distinct colours. The following saturation problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger is considered in Chapter~$5$. Given a graph $H$ and a set of colours $\{1,2,\ldots,t\}$ (for some integer $t\geq |E(H)|$), we define $sat_{t}(n, R(H))$ to be the minimum number of $t$-coloured edges in a graph on $n$ vertices which does not contain a rainbow copy of $H$ but the addition of any non-edge in any colour from $\{1,2,\ldots,t\}$ creates such a copy. We prove several results concerning these extremal numbers. In particular, we determine the correct order of $sat_{t}(n, R(H))$, as a function of $n$, for every connected graph $H$ of minimum degree greater than $1$ and for every integer $t\geq e(H)$. In Chapter~$6$, we consider the following question: under what conditions does a Hamiltonian graph on $n$ vertices possess a second cycle of length at least $n-o(n)$? We prove that the `weak' assumption of a minimum degree greater or equal to $3$ guarantees the existence of such a long cycle. We solve two problems related to majority colouring in Chapter~$7$. This topic was recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number $k$, the smallest positive integer $m = m(k)$ such that every digraph can be coloured with $m$ colours, where each vertex has the same colour as at most a proportion of $\frac{1}{k}$ of its out-neighbours. Our main theorem states that $m(k) \in \{2k-1, 2k\}$. We study the following problem, raised by Caro and Yuster, in Chapter~$8$. Does every graph $G$ contain a `large' induced subgraph $H$ which has $k$ vertices of degree exactly $\Delta(H)$? We answer in the affirmative an approximate version of this question. Indeed, we prove that, for every $k$, there exists $g(k)$ such that any $n$ vertex graph $G$ with maximum degree $\Delta$ contains an induced subgraph $H$ with at least $n-g(k)\sqrt{\Delta}$ vertices such that $V(H)$ contains at least $k$ vertices of the same degree $d \ge \Delta(H)-g(k)$. This result is sharp up to the order of $g(k)$. %Subsequently, we investigate a concept called $\textit{path-pairability}$. A graph is said to be path-pairable if for any pairing of its vertices there exist a collection of edge-disjoint paths routing the the vertices of each pair. A question we are concerned here asks whether every planar path pairable graph on $n$ vertices must possess a vertex of degree linear in $n$. Indeed, we answer this question in the affirmative. We also sketch a proof resolving an analogous question for graphs embeddable on surfaces of bounded genus. Finally, in Chapter~$9$, we move on to examine $k$-linked tournaments. A tournament $T$ is said to be $k$-linked if for any two disjoint sets of vertices $\{x_1,\ldots ,x_k\}$ and $\{y_1,\dots,y_k\}$ there are directed vertex disjoint paths $P_1,\dots, P_k$ such that $P_i$ joins $x_i$ to $y_i$ for $i = 1,\ldots, k$. We prove that any $4k$ strongly-connected tournament with sufficiently large minimum out-degree is $k$-linked. This result comes close to proving a conjecture of Pokrovskiy

Probabilistic methods for distributed information dissemination

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2013.Cataloged from PDF version of thesis.Includes bibliographical references (p. 457-484).The ever-increasing growth of modern networks comes with a paradigm shift in network operation. Networks can no longer be abstracted as deterministic, centrally controlled systems with static topologies but need to be understood as highly distributed, dynamic systems with inherent unreliabilities. This makes many communication, coordination and computation tasks challenging and in many scenarios communication becomes a crucial bottleneck. In this thesis, we develop new algorithms and techniques to address these challenges. In particular we concentrate on broadcast and information dissemination tasks and introduce novel ideas on how randomization can lead to powerful, simple and practical communication primitives suitable for these modern networks. In this endeavor we combine and further develop tools from different disciplines trying to simultaneously addresses the distributed, information theoretic and algorithmic aspects of network communication. The two main probabilistic techniques developed to disseminate information in a network are gossip and random linear network coding. Gossip is an alternative to classical flooding approaches: Instead of nodes repeatedly forwarding information to all their neighbors, gossiping nodes forward information only to a small number of (random) neighbors. We show that, when done right, gossip disperses information almost as quickly as flooding, albeit with a drastically reduced communication overhead. Random linear network coding (RLNC) applies when a large amount of information or many messages are to be disseminated. Instead of routing messages through intermediate nodes, that is, following a classical store-and-forward approach, RLNC mixes messages together by forwarding random linear combinations of messages. The simplicity and topology-obliviousness of this approach makes RLNC particularly interesting for the distributed settings considered in this thesis. Unfortunately the performance of RLNC was not well understood even for the simplest such settings. We introduce a simple yet powerful analysis technique that allows us to prove optimal performance guarantees for all settings considered in the literature and many more that were not analyzable so far. Specifically, we give many new results for RLNC gossip algorithms, RLNC algorithms for dynamic networks, and RLNC with correlated data. We also provide a novel highly efficient distributed implementation of RLNC that achieves these performance guarantees while buffering only a minimal amount of information at intermediate nodes. We then apply our techniques to improve communication primitives in multi-hop radio networks. While radio networks inherently support broadcast communications, e.g., from one node to all surrounding nodes, interference of simultaneous transmissions makes multihop broadcast communication an interesting challenge. We show that, again, randomization holds the key for obtaining simple, efficient and distributed information dissemination protocols. In particular, using random back-off strategies to coordinate access to the shared medium leads to optimal gossip-like communications and applying RLNC achieves the first throughput-optimal multi-message communication primitives. Lastly we apply our probabilistic approach for analyzing simple, distributed propagation protocols in a broader context by studying algorithms for the Lovász Local Lemma. These algorithms find solutions to certain local constraint satisfaction problems by randomly fixing and propagating violations locally. Our two main results show that, firstly, there are also efficient deterministic propagation strategies achieving the same and, secondly, using the random fixing strategy has the advantage of producing not just an arbitrary solution but an approximately uniformly random one. Both results lead to simple, constructions for a many locally consistent structures of interest that were not known to be efficiently constructable before.by Bernhard Haeupler.Ph.D

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Utilisation de l'espace par le raton laveur et la moufette rayée, deux principaux hôtes d'un variant du virus de la rage

Comprendre les mécanismes comportementaux qui régissent la transmission et la propagation de pathogènes est critique pour les programmes de contrôle et de prévention des maladies infectieuses. Ma thèse explore le lien entre l’hétérogénéité des paysages agroforestiers et la répartition spatio-temporelle du raton laveur (Procyon lotor) et de la moufette rayée (Mephitis mephitis), deux principaux hôtes du variant de la rage du raton laveur. Pour cela, j’ai étudié les processus de sélection densité-dépendante de l’habitat à large et fine échelles spatiales, ainsi que les mécanismes de connectivité fonctionnelle chez ces deux espèces. Le chapitre 1 a révélé qu’à faibles abondances de congénères, les ratons laveurs sélectionnaient les secteurs avec une forte proportion de forêts, tandis que les moufettes rayées préféraient les zones avec une large proportion de milieux anthropiques. À fortes abondances, cependant, les deux espèces sélectionnaient plutôt les secteurs composés d’une forte densité de bordures maïs-forêts et forte proportion de champs de maïs. Le chapitre 2 a montré que la sélection des champs de maïs par les ratons laveurs dépendait à la fois de la densité de congénères et de l’abondance des champs de maïs. Dans le chapitre 3, j’ai développé un modèle spatialement explicite basé sur l’individu pour évaluer comment des règles empiriques de déplacement font émerger des patrons de contacts chez des individus hôtes. Les simulations ont révélé trois patrons généraux dans les taux de contacts. Premièrement, une petite portion de ratons laveurs simulés était responsable de la majorité des contacts dans les paysages virtuels. Deuxièmement, les taux de contacts des ratons laveurs simulés augmentaient linéairement avec la densité de congénères plutôt qu’avec la proportion de congénères dans la plupart des paysages. Troisièmement, l’effet de la connectivité fonctionnelle sur les taux de contacts variait fortement en fonction des types de milieux et de leur disponibilité dans le paysage. Les modèles développés dans cette thèse procurent une base solide au développement de programmes de contrôle et de prévention des maladies infectieuses, en permettant d’identifier les zones à hautes densités d’individus et de taux de contacts entre eux, et donc où le risque de transmission de pathogènes est relativement élevé.Understanding behavioral mechanisms that determine the transmission and spread of pathogens is critical for control and prevention programs of infectious diseases. My thesis investigates the interplay between the heterogeneity of agriculturally fragmented landscapes and spatio-temporal distribution patterns of raccoons (Procyon lotor) and striped skunks (Mephitis mephitis), two main hosts of the raccoon rabies virus variant. To do this, I studied the processes of density-dependent habitat selection at large and fine spatial scales, together with the mechanisms determining functional connectivity for these two species. The first chapter revealed that at low conspecific abundances in the landscape, raccoons selected areas with a high proportion of forests, whereas striped skunks preferred areas with a large proportion of anthropogenic features. At high conspecific abundances, however, both species rather selected areas composed of a high density of corn-forest edges and a large proportion of corn fields. The second chapter showed that raccoons altered their selection of corn fields depending upon both conspecific density and abundance of corn fields. In the third chapter, I built a spatially explicit individual-based model to assess how empirical movement rules translate into spatial patterns of contact rates among individual hosts. The simulations revealed three general patterns in contact rates. First, a small number of simulated raccoons were responsible for the majority of contacts in virtual landscapes. Second, contact rates of simulated raccoons increased linearly with conspecific density rather than with the proportion of conspecifics in most of the virtual landscapes. Third, the influence of functional connectivity on contact rates varied strongly among land cover types and with their availability in the landscape. The models developed in this thesis provide a strong basis upon which to build control and prevention programs for infectious diseases, as they identify areas where animal density and contact rates should be relatively high and, hence, where the risk of pathogen transmission should also be high