In , Nguyen and Onak constructed the first constant-time algorithm for the approximation of the size of the maximum matching in bounded degree graphs. The Borel oracle machinery is a tool that can be used to convert some statements in Borel graph theory to theorems in the field of constant-time algorithms. In this paper we illustrate the power of this… (More)
We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes' Embedding Conjecture as well as the Measurable Determinant Conjecture of Lück, Sauer and Wegner hold for treeable equivalence relations.
We show that a sufficiently large graph of bounded degree can be decomposed into quasi-homogeneous pieces in a canonical way. The result can be viewed as a " finitarization " of the classical Farrell-Varadarajan Ergodic Decomposition Theorem.
The classical theorem of Vizing states that every graph of maximum degree d admits an edge-coloring with at most d + 1 colors. Furthermore, as it was earlier shown by K˝ onig, d colors suffice if the graph is bipartite. We investigate the existence of measurable edge-colorings for graphings. A graphing is an analytic generalization of a bounded-degree graph… (More)
We study " positive " graphs that have a nonnegative homomor-phism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary simple graph and gluing them together along an independent set of nodes. We prove the conjecture for various classes of… (More)