Related papers: Sparse String Graphs and Region Intersection Graph…
The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic questions. In this paper we…
We conjecture that finite graphs with positive Cheeger constant admit a spanning subgraph with positive Cheeger constant and girth proportional to the diameter. We prove this conjecture for regular expander graphs with large expansion. Our…
We propose and investigate a unifying class of sparse random graph models, based on a hidden coloring of edge-vertex incidences, extending an existing approach, Random graphs with a given degree distribution, in a way that admits a…
An intersection graph of curves in the plane is called a string graph. Matousek almost completely settled a conjecture of the authors by showing that every string graph of m edges admits a vertex separator of size O(\sqrt{m}\log m). In the…
Constructing a spanning tree of a graph is one of the most basic tasks in graph theory. We consider a relaxed version of this problem in the setting of local algorithms. The relaxation is that the constructed subgraph is a sparse spanning…
We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient…
Colouring sparse graphs under various restrictions is a theoretical problem of significant practical relevance. Here we consider the problem of maximizing the number of different colours available at the nodes and their neighbourhoods,…
In this paper, we study unique colourings in random graphs as a generalization of both conflict-free and injective colourings. Specifically, we impose the condition that a fraction of vertices in the neighbourhood of any vertex are assigned…
A family of sets in the plane is simple if the intersection of its any subfamily is arc-connected, and it is pierced by a line $L$ if the intersection of its any member with $L$ is a nonempty segment. It is proved that the intersection…
We prove that any bounded degree regular graph with sufficiently strong spectral expansion contains an induced path of linear length. This is the first such result for expanders, strengthening an analogous result in the random setting by…
Weak and strong coloring numbers are generalizations of the degeneracy of a graph, where for each natural number $k$, we seek a vertex ordering such every vertex can (weakly respectively strongly) reach in $k$ steps only few vertices with…
We survey work on coloring, list coloring, and painting squares of graphs; in particular, we consider strong edge-coloring. We focus primarily on planar graphs and other sparse classes of graphs.
A graph coloring has bounded clustering if each monochromatic component has bounded size. This paper studies such a coloring, where the number of colors depends on an excluded complete bipartite subgraph. This is a much weaker assumption…
We derive sharp upper and lower bounds on the number of intersection points and closed regions that can occur in sets of line segments with certain structure, in terms of the number of segments. We consider sets of segments whose underlying…
The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for…
We consider the problem of stretching pseudolines in a planar straight-line drawing to straight lines while preserving the straightness and the combinatorial embedding of the drawing. We answer open questions by Mchedlidze et al. by showing…
We investigate in some detail a recently suggested general class of ensembles of sparse undirected random graphs based on a hidden stub-coloring, with or without the restriction to nondegenerate graphs. The calculability of local and global…
Some old and new evidence for the existence of the string (planar random surfaces) representation of multicolour QCD are reviewed. They concern the random surface representation of the strong coupling expansion in lattice multicolour gauge…
We revisit the classical question of the relationship between the diameter of a graph and its expansion properties. One direction is well understood: expander graphs exhibit essentially the lowest possible diameter. We focus on the reverse…
We investigate string graphs through the lens of graph product structure theory, which describes complicated graphs as subgraphs of strong products of simpler building blocks. A graph $G$ is called a string graph if its vertices can be…