Related papers: Borel line graphs

Let $G$ be a simple finite connected graph. The line graph $L(G)$ of graph $G$ is the graph whose vertices are the edges of $G$, where $ef \in E(L(G))$ when $e \cap f \neq \emptyset$. Iteratively, the higher order line graphs are defined…

We construct, for each countable ordinal $\xi$, a closed graph with Borel chromatic number two and Baire class $\xi$ chromatic number $\aleph\_0$.

We investigate when a Borel graph admits a (Borel or measurable) orientation with outdegree bounded by $k$ for various cardinals $k$. We show that for a p.m.p. graph $G$, a measurable orientation can be found when $k$ is larger than the…

We consider the graph class Grounded-L corresponding to graphs that admit an intersection representation by L-shaped curves, where additionally the topmost points of each curve are assumed to belong to a common horizontal line. We prove…

We consider a large family of theories of equivalence relations, each with finitely many classes, and assuming the existence of an $\omega$-Erdos cardinal, we determine which of these theories are Borel complete. We develop machinery,…

We construct bounded degree acyclic Borel graphs with large Borel chromatic number using a graph arising from Ramsey theory and limits of expander sequences.

We show that if a locally finite Borel graph with quasitransitive amenable components admits a fractional perfect matching, it will admit a Borel fractional perfect matching. In particular, if a countable amenable quasitransitive graph…

We demonstrate that the proper homotopy equivalence relation for locally finite graphs is Borel complete. Furthermore, among the infinite graphs, there is a comeager equivalence class. As corollaries, we obtain the analogous results for the…

We show that the set of locally finite Borel graphs with finite Borel asymptotic dimension is $\mathbf{\Sigma}^1_2$-complete. The result is based on a combinatorial characterization of finite Borel asymptotic dimension for graphs generated…

In this paper we consider coloring problems on graphs and other combinatorial structures on standard Borel spaces. Our goal is to obtain sufficient conditions under which such colorings can be made well-behaved in the sense of topology or…

We show that a locally finite Borel graph is nonsmooth if and only if it admits marker sequences which are "far" from every point. Our proof uses the Galvin-Prikry theorem and the Glimm-Effros dichotomy.

We prove a descriptive version of Brooks's theorem for directed graphs. In particular, we show that, if $D$ is a Borel directed graph on a standard Borel space $X$ such that the maximum degree of each vertex is at most $d \geq 3$, then…

This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence…

Let $\mathbf G$ be a graphing, that is a Borel graph defined by $d$ measure preserving involutions. We prove that if $\mathbf G$ is {\em treeable} then it arises as the local limit of some sequence $(G_n)_{n\in\mathbb{N}}$ of graphs with…

Bond percolation on Cayley graphs provides examples of random graphs. Other examples arise from the dynamical study of proper repetitive subgraphs of Cayley graphs. In this paper we demonstrate that these two families have mutually singular…

For each commutative, graded algebra with finite dimension in each degree, we construct a graded cohomology theory for graphs whose graded Euler characteristic is the chromatic polynomial of the graph. This extends our previous work which…

Let k be a natural number. We introduce k-threshold graphs. We show that there exists an O(n^3) algorithm for the recognition of k-threshold graphs for each natural number k. k-Threshold graphs are characterized by a finite collection of…

We prove a full measurable version of Vizing's theorem for bounded degree Borel graphs, that is, we show that every Borel graph $\mathcal{G}$ of degree uniformly bounded by $\Delta\in \mathbb{N}$ defined on a standard probability space…

We show that every $d$-regular bipartite Borel graph admits a Baire measurable $k$-regular spanning subgraph if and only if $d$ is odd or $k$ is even. This gives the first example of a locally checkable coloring problem which is known to…

We study the analytic digraphs of uncountable Borel chromatic number on Polish spaces, and compare them with the notion of injective Borel homomorphism. We provide some minimal digraphs incomparable with G 0. We also prove the existence of…