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Using the spectral theory of weakly convergent sequences of finite graphs, we prove the uniform existence of the integrated density of states for a large class of infinite graphs.
This paper introduces a new graph construction, the permutational power of a graph, whose adjacency matrix is obtained by the composition of a permutation matrix with the adjacency matrix of the graph. It is shown that this construction…
In this note, we prove an interesting result about perfect matchings in a complete bipartite graph with 2n vertices on each side, whose edges are colored in red and blue such that each vertex is part of n red edges and n blue edges.
Let $G$ be a graph that admits a perfect matching. A {\sf forcing set} for a perfect matching $M$ of $G$ is a subset $S$ of $M$, such that $S$ is contained in no other perfect matching of $G$. This notion originally arose in chemistry in…
We consider properties of edge-colored vertex-ordered graphs, i.e., graphs with a totally ordered vertex set and a finite set of possible edge colors. We show that any hereditary property of such graphs is strongly testable, i.e., testable…
Binomial Cayley graphs are obtained by considering the binomial coefficient of the weight function of a given Cayley graph and a natural number. We introduce these objects and study two families: one associated with symmetric groups and the…
Beck's conjecture on coloring of graphs associated to various algebraic objects has generated considerable interest in the community of discrete mathematics and combinatorics since its inception in the year 1988. The version of this…
It has previously been hypothesized, and supported with some experimental evidence, that deeper representations, when well trained, tend to do a better job at disentangling the underlying factors of variation. We study the following related…
The 3-coloring of hereditary graph classes has been a deeply-researched problem in the last decade. A hereditary graph class is characterized by a (possibly infinite) list of minimal forbidden induced subgraphs $H_1,H_2,\ldots$; the graphs…
We revisit the problem of enumeration of vertex-tricolored planar random triangulations solved in [Nucl. Phys. B 516 [FS] (1998) 543-587] in the light of recent combinatorial developments relating classical planar graph counting problems to…
We prove several consistency results in choiceless set theory ZF+DC regarding countable chromatic numbers of various algebraic hypergraphs on Euclidean spaces.
We study the approximability of an existing framework for clustering edge-colored hypergraphs, which is closely related to chromatic correlation clustering and is motivated by machine learning and data mining applications where the goal is…
We give a characterization of finite sets of triples of elements (e.g., positive integers) that can be colored with two colors such that for every element $i$ in each color class there exists a triple which does not contain $i$. We give a…
Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that…
Given an undirected graph $G=(V,E)$ with a set of vertices $V$ and a set of edges $E$, a graph coloring problem involves finding a partition of the vertices into different independent sets. In this paper we present a new framework that…
Color refinement is a classical technique used to show that two given graphs G and H are non-isomorphic; it is very efficient, although it does not succeed on all graphs. We call a graph G amenable to color refinement if it succeeds in…
The chromatic polynomials are studied by several authors and have important applications in different frameworks, specially, in graph theory and enumerative combinatorics. The aim of this work is to establish some properties of the…
Graph-structured data appears frequently in domains including chemistry, natural language semantics, social networks, and knowledge bases. In this work, we study feature learning techniques for graph-structured inputs. Our starting point is…
Let $X$ be a (repetitive) infinite connected simple graph with a finite upper bound $\Delta$ on the vertex degrees. The main theorem states that $X$ admits a (repetitive) limit aperiodic vertex coloring by $\Delta$ colors. This refines a…
The study of sorting permutations by block interchanges has recently been stimulated by a phenomenon observed in the genome maintenance of certain ciliate species. The result was the identification of a block interchange operation that…