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We study weighted edge coloring of graphs, where we are given an undirected edge-weighted general multi-graph $G := (V, E)$ with weights $w : E \rightarrow [0, 1]$. The goal is to find a proper weighted coloring of the edges with as few…
Neutral atom arrays have emerged as a versatile candidate for the embedding of hard classical optimization problems. Prior work has focused on mapping problems onto finding the maximum independent set of weighted or unweighted unit disk…
We study the graph alignment problem over two independent Erd\H{o}s-R\'enyi graphs on $n$ vertices, with edge density $p$ falling into two regimes separated by the critical window around $p_c=\sqrt{\log n/n}$. Our result reveals an…
A coloring of edges of a finite directed graph turns the graph into finite-state automaton. The synchronizing word of a deterministic automaton is a word in the alphabet of colors (considered as letters) of its edges that maps the automaton…
We study a graph coloring problem that is otherwise easy but becomes quite non-trivial in the one-pass streaming model. In contrast to previous graph coloring problems in streaming that try to find an assignment of colors to vertices, our…
The chosen tool of this thesis is an extremal type approach. The lesson drawn by the theorems proved in the thesis is that surprisingly small compromise is necessary on the efficacy of the solutions to make the approach work. The problems…
This paper deals with the complexity of some natural graph problems when parametrized by {measures that are restrictions of} clique-width, such as modular-width and neighborhood diversity. The main contribution of this paper is to introduce…
We propose quasi-stable coloring, an approximate version of stable coloring. Stable coloring, also called color refinement, is a well-studied technique in graph theory for classifying vertices, which can be used to build compact, lossless…
An introduction to the application of combinatorial optimization methods to ground state calculations of frustrated, disordered systems is given. We discuss the interface problem in the random bond Ising ferromagnet, the random field Ising…
Let $G$ be an $n$-vertex graph with the maximum degree $\Delta$ and the minimum degree $\delta$. We give algorithms with complexity $O(1.3158^{n-0.7~\Delta(G)})$ and $O(1.32^{n-0.73~\Delta(G)})$ that determines if $G$ is 3-colorable, when…
We consider entropy-optimal graphons associated with extreme and near-extreme constraints on the densities of edges and triangles. We prove that the optimizers for near-extreme constraints are unique and multipodal and are perturbations of…
The design of a good algorithm to solve NP-hard combinatorial approximation problems requires specific domain knowledge about the problems and often needs a trial-and-error problem solving approach. Graph coloring is one of the essential…
Given a graph $G$ that is modified by a sequence of edge insertions and deletions, we study the Maximum $k$-Edge Coloring problem Having access to $k$ colors, how can we color as many edges of $G$ as possible such that no two adjacent edges…
We introduce a generalization of the well known graph (vertex) coloring problem, which we call the problem of \emph{component coloring of graphs}. Given a graph, the problem is to color the vertices using minimum number of colors so that…
This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so…
Graph colouring is a fundamental problem for networks, serving as a tool for avoiding conflicts via symmetry breaking, for example, avoiding multiple computer processes simultaneously updating the same resource. This paper considers a…
We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the "inhomogeneous random graph model" of S\"oderberg (2002), extended by Bollob\'as, Janson and Riordan (2007). By means of analytic…
This paper explores the application of a new algebraic method of color exchanges to the edge coloring of simple graphs. Vizing's theorem states that the edge coloring of a simple graph $G$ requires either $\Delta$ or $\Delta+1$ colors,…
We describe a general-purpose method for finding high-quality solutions to hard optimization problems, inspired by self-organized critical models of co-evolution such as the Bak-Sneppen model. The method, called Extremal Optimization,…
The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a…