Generalized Colorings of Graphs

A graph coloring is an assignment of labels called “colors” to certain elements of a graph subject to certain constraints. The proper vertex coloring is the most common type of graph coloring, where each vertex of a graph is assigned one color such that no two adjacent vertices share the same color, with the objective of minimizing the number of colors used. One can obtain various generalizations of the proper vertex coloring problem, by strengthening or relaxing the constraints or changing the objective. We study several types of such generalizations in this thesis. Series-parallel graphs are multigraphs that have no K4-minor. We provide bounds on their fractional and circular chromatic numbers and the defective version of these pa-rameters. In particular we show that the fractional chromatic number of any series-parallel graph of odd girth k is exactly 2k/(k − 1), confirming a conjecture by Wang and Yu. We introduce a generalization of defective coloring: each vertex of a graph is assigned a fraction of each color, with the total amount of colors at each vertex summing to 1. We define the fractional defect of a vertex v to be the sum of the overlaps with each neighbor of v, and the fractional defect of the graph to be the maximum of the defects over all vertices. We provide results on the minimum fractional defect of 2-colorings of some graphs. We also propose some open questions and conjectures. Given a (not necessarily proper) vertex coloring of a graph, a subgraph is called rainbow if all its vertices receive different colors, and monochromatic if all its vertices receive the same color. We consider several types of coloring here: a no-rainbow-F coloring of G is a coloring of the vertices of G without rainbow subgraph isomorphic to F ; an F -WORM coloring of G is a coloring of the vertices of G without rainbow or monochromatic subgraph isomorphic to F ; an (M, R)-WORM coloring of G is a coloring of the vertices of G with neither a monochromatic subgraph isomorphic to M nor a rainbow subgraph isomorphic to R. We present some results on these concepts especially with regards to the existence of colorings, complexity, and optimization within certain graph classes. Our focus is on the case that F , M or R is a path, cycle, star, or clique

The Chromatic Structure of Dense Graphs

This thesis focusses on extremal graph theory, the study of how local constraints on a graph affect its macroscopic structure. We primarily consider the chromatic structure: whether a graph has or is close to having some (low) chromatic number. Chapter 2 is the slight exception. We consider an induced version of the classical Turán problem. Introduced by Loh, Tait, Timmons, and Zhou, the induced Turán number ex(n, {H, F-ind}) is the greatest number of edges in an n-vertex graph with no copy of H and no induced copy of F. We asymptotically determine ex(n, {H, F-ind}) for H not bipartite and F neither an independent set nor a complete bipartite graph. We also improve the upper bound for ex(n, {H, K_{2, t}-ind}) as well as the lower bound for the clique number of graphs that have some fixed edge density and no induced K_{2, t}. The next three chapters form the heart of the thesis. Chapters 3 and 4 consider the Erdős-Simonovits question for locally r-colourable graphs: what are the structure and chromatic number of graphs with large minimum degree and where every neighbourhood is r-colourable? Chapter 3 deals with the locally bipartite case and Chapter 4 with the general case. While the subject of Chapters 3 and 4 is a natural local to global colouring question, it is also essential for determining the minimum degree stability of H-free graphs, the focus of Chapter 5. Given a graph H of chromatic number r + 1, this asks for the minimum degree that guarantees that an H-free graph is close to r-partite. This is analogous to the classical edge stability of Erdős and Simonovits. We also consider the question for the family of graphs to which H is not homomorphic, showing that it has the same answer. Chapter 6 considers sparse analogues of the results of Chapters 3 to 5 obtaining the thresholds at which the sparse problem degenerates away from the dense one. Finally, Chapter 7 considers a chromatic Ramsey problem first posed by Erdős: what is the greatest chromatic number of a triangle-free graph on $n$ vertices or with m edges? We improve the best known bounds and obtain tight (up to a constant factor) bounds for the list chromatic number, answering a question of Cames van Batenburg, de Joannis de Verclos, Kang, and Pirot

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Certification of many-body systems

Quantum physics is arguably both the most successful and the most counterintuitive physical theory of all times. Its extremely accurate predictions on the behaviour of microscopic particles have led to unprecedented technological advances in various fields and yet, many quantum phenomena defy our classical intuition. Starting from the 1980’s, however, a paradigm shift has gradually taken hold in the scientific community, consisting in studying quantum phenomena not as inexplicable conundrums but as useful resources. This shift marked the birth of the field of quantum information science, which has since then explored the advantages that quantum theory can bring to the way we process and transfer information. In this thesis, we introduce scalable certification tools that apply to various operational properties of many-body quantum systems. In the first three cases we consider, we base our certification protocols on the detection of nonlocal correlations. These kinds of non-classical correlations that can displayed by quantum states allow one to assess relevant properties in a device-independent manner, that is, without assuming anything about the specific functioning of the device producing the state of interest or the implemented measurements. In the first scenario we present an efficient method to detect multipartite entanglement in a device-independent way. We do so by introducing a numerical test for nonlocal correlations that involves computational and experimental resources that scale polynomially with the system number of particles. We show the range of applicability of the method by using it to detect entanglement in various families of multipartite systems. In multipartite systems, however, it is often more informative to provide quantitative statements. We address this problem in the second scenario by introducing scalable methods to quantify the nonlocality depth of a multipartite systems, that is, the number of particles sharing nonlocal correlations among each other. We show how to do that by making use of the knowledge of two-body correlations only and we apply the resulting techniques to experimental data from a system of a few hundreds of atoms. In the third scenario, we move to consider self-testing, which is the most informative certification method based on nonlocality. Indeed, in a self-testing task, one is interested in characterising the state of the system and the measurement performed on it, by simply looking at the resulting correlations. We introduce the first scalable self-testing method based on Bell inequalities and apply it to graph states, a well-known family of multipartite quantum states. Moreover, we show that the certification achieved with our method is robust against experimental imperfections. Lastly, we address the problem of certifying the result of quantum optimizers. They are quantum devices designed to estimate the groundstate energy of classical spin systems. We provide a way to efficiently compute a convergent series of upper and lower bounds to the minimum of interest, which at each step allows one to certify the output of any quantum optimizer.La física cuántica es posiblemente la teoría física más exitosa y la más contraintuitiva jamás desarollada. A pesar de que sus predicciones extremadamente precisas sobre el comportamiento de las partículas microscópicas han llevado a avances tecnológicos sin precedentes en varios campos, muchos fenómenos cuánticos desafían nuestra intuición basada en una concepción clásica de la física. Sin embargo, a partir de la década de 1980 tuvo lugar un cambio de paradigma en la comunidad científica, que se orientó en estudiar los fenómenos cuánticos no como enigmas inexplicables, sino como recursos útiles. Este cambio marcó el nacimiento del campo de la ciencia de la información cuántica, que desde entonces ha explorado las ventajas que la teoría cuántica puede aportar a la forma en que procesamos y transferimos la información. Hoy en día es un hecho bien establecido que la codificación de información en partículas cuánticas puede llevar, por ejemplo, a procesos de cálculo más eficientes, así como a comunicaciones extremadamente seguras. Además, debido a sus aplicaciones prácticas a la vida cotidiana, la ciencia de la información cuántica ha atraído un gran interés político y económico. Recientemente se han lanzado varias iniciativas con el propósito de cerrar la brecha entre la ciencia básica y la industria en este campo, tanto a nivel nacional como internacional. Al mismo tiempo, cada vez más empresas están incrementando sus esfuerzos para producir dispositivos cuánticos a nivel comercial. No hay duda de que hemos entrado en la era de la primera generación de dispositivos cuánticos, en la cual los sistemas cuánticos controlables compuestos de decenas o cientos de partículas son cada vez más accesibles. En tal escenario, el certificar que estos dispositivos exhiben sus atractivas propiedades cuánticas constituye un problema fundamental. Es importante destacar que, para que los métodos de certificación deseados sean aplicables en situaciones reales, éstos deben ser escalables con el tamaño del sistema. En otras palabras, tienen que basarse en requerimientos computacionales y experimentales que crezcan, a lo sumo,polinomialmente con el número de partículas en el sistema de interés. En esta tesis, introducimos herramientas de certificación escalables que se aplican a varias propiedades operativas de sistemas cuánticos de muchos cuerpos. En los primeros tres casos que consideramos, basamos nuestros protocolos de certificación en la detección de correlaciones no locales. Estos tipos de correlaciones no clásicas, que únicamente pueden ser producidas por sistemas cuánticos, permiten evaluar propiedades relevantes de forma independiente del dispositivo, es decir, sin realizar hipótesis acerca del funcionamiento específico del dispositivo que produce el estado de interés o las mediciones implementadas. En el primer escenario, presentamos un método eficiente para detectar entrelazamiento en sistemas multipartitos de forma independiente del dispositivo. Lo hacemos mediante la introducción de una prueba numérica para las correlaciones no locales que involucra recursos computacionales y experimentales que escalan polinomialmente con el número de partículas del sistema. Mostramos el rango de aplicabilidad de dicho método usándolo para detectar entrelazamiento en varias familias de sistemas multipartitos. Sin embargo, al tratar con sistemas de muchos cuerpos a menudo es más informativo proporcionar informaciones cuantitativas. Abordamos este problema en el segundo escenario mediante la introducción de métodos escalables para cuantificar la profundidad no local (non-locality depth) de un sistema multipartito, es decir, la cantidad de partículas que comparten correlaciones no locales entre sí. Mostramos cómo realizar dicha cuantificación a partir del conocimiento únicamente de los correladores de dos cuerpos, y aplicamos las técnicas resultantes a los datos experimentales de un sistema de unos pocos cientos de átomos. En el tercer escenario, pasamos a considerar el caso de self-testing, que es el método de certificación más informativo basado en la no localidad. De hecho, en una tarea de self-testing, el objetivo es caracterizar el estado del sistema y las mediciones realizadas en él, simplemente observando las correlaciones resultantes. Introducimos el primer método de self-testing escalable basado en las desigualdades de Bell y lo aplicamos a estados de grafo, una familia muy conocida de estados cuánticos multipartitos. Además, demostramos que la certificación lograda con nuestro método es robusta a imperfecciones experimentales. Por último, consideramos el problema de certificar el resultado de optimizadores cuánticos. Estos son dispositivos cuánticos diseñados para estimar la energía del estado fundamental de sistemas de espines clásicos. Desarollamos un método eficiente para calcular una serie convergente de límites superiores e inferiores al mínimo de interés, que en cada paso permite certificar el resultado de cualquier optimizador cuántic