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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…
Graph colorings is a fundamental topic in graph theory that require an assignment of labels (or colors) to vertices or edges subject to various constraints. We focus on the harmonious coloring of a graph, which is a proper vertex coloring…
Using the algebraic approach to promise constraint satisfaction problems, we establish complexity classifications of three natural variants of hypergraph colourings: standard nonmonochromatic colourings, conflict-free colourings, and…
In this paper, from a theoretical perspective, we study how powerful graph neural networks (GNNs) can be for learning approximation algorithms for combinatorial problems. To this end, we first establish a new class of GNNs that can solve a…
Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is…
Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic…
In this work, we present a branch-and-price algorithm to solve the weighted version of the List Coloring Problem, based on a vertex cover formulation by stable sets. This problem is interesting for its applications and also for the many…
A classic graph coloring problem is to assign colors to vertices of any graph so that distinct colors are assigned to adjacent vertices. Optimal graph coloring colors a graph with a minimum number of colors, which is its chromatic number.…
The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In…
We present a proof, using spectral techniques, that there is no finite measurable coloring of the odd-distance graph.
We give a near-linear time 4-coloring algorithm for planar graphs, improving on the previous quadratic time algorithm by Robertson et al. from 1996. Such an algorithm cannot be achieved by the known proofs of the Four Color Theorem (4CT).…
We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the…
Recent developments in approximate counting have made startling progress in developing fast algorithmic methods for approximating the number of solutions to constraint satisfaction problems (CSPs) with large arities, using connections to…
Many complex systems and datasets are characterized by multiway interactions of different categories, and can be modeled as edge-colored hypergraphs. We focus on clustering such datasets using the NP-hard edge-colored clustering problem,…
In this paper, we study the conflict-free coloring of graphs induced by neighborhoods. A coloring of a graph is conflict-free if every vertex has a uniquely colored vertex in its neighborhood. The conflict-free coloring problem is to color…
In the Coloured Clustering problem, we wish to colour vertices of an edge coloured graph to produce as many stable edges as possible, i.e., edges with the same colour as their ends. In this paper, we reveal that the problem is in fact a…
Deep learning has consistently defied state-of-the-art techniques in many fields over the last decade. However, we are just beginning to understand the capabilities of neural learning in symbolic domains. Deep learning architectures that…
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…
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a…
Colour refinement is a basic algorithmic routine for graph isomorphism testing, appearing as a subroutine in almost all practical isomorphism solvers. It partitions the vertices of a graph into "colour classes" in such a way that all…