Related papers: Approximate Schreier decorations and approximate K…

Hajnal and Szemer\'{e}di proved that if $G$ is a finite graph with maximum degree $\Delta$, then for every integer $k \geqslant \Delta+1$, $G$ has a proper coloring with $k$ colors in which every two color classes differ in size at most by…

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…

Let $X$ be a Polish space with Borel probability measure $\mu,$ and let $G$ be a Borel graph on $X$ with no odd cycles and maximum degree $\Delta(G).$ We show that the Baire measurable edge chromatic number of $G$ is at most $\Delta(G)+1$,…

We work with simple graphs in ZF (Zermelo--Fraenkel set theory without the Axiom of Choice (AC)) and assume that the sets of colors can be either well-orderable or non-well-orderable to prove that the following statements are equivalent to…

We establish two versions of Vizing's theorem for Borel multi-graphs whose vertex degrees and edge multiplicities are uniformly bounded by respectively $\Delta$ and $\pi$. The ``approximate'' version states that, for any Borel probability…

We show that every Borel graph $G$ of subexponential growth has a Borel proper edge-coloring with $\Delta(G) + 1$ colors. We deduce this from a stronger result, namely that an $n$-vertex (finite) graph $G$ of subexponential growth can be…

We generalize Brooks's theorem to show that if $G$ is a Borel graph on a standard Borel space $X$ of degree bounded by $d \geq 3$ which contains no $(d+1)$-cliques, then $G$ admits a $\mu$-measurable $d$-coloring with respect to any Borel…

In an unfriendly coloring of a graph the color of every node mismatches that of the majority of its neighbors. We show that every probability measure preserving Borel graph with finite average degree admits a Borel unfriendly coloring…

Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number…

The Gr\"{o}tzsch Theorem states that every triangle-free planar graph admits a proper $3$-coloring. Among many of its generalizations, the one of Gr\"{u}nbaum and Aksenov, giving $3$-colorability of planar graphs with at most three…

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…

We prove that every $2d$-regular unimodular random network carries an invariant random Schreier decoration. Equivalently, it is the Schreier coset graph of an invariant random subgroup of the free group $F_d$. As a corollary we get that…

In this paper we consider the Borel combinatorics of Schreier graphs of $\mathbb{Z}$-actions with arbitrary finite generating sets. We formulate the Borel combinatorics in terms of existence of Borel equivariant maps from…

We initiate a systematic study of spectral theory for bounded-degree Borel pmp graphs. Specifically, we study spectral properties of the associated adjacency and Laplacian operators. We start with proving a spectral characterization of…

If $L$ is a list assignment of $r$ colors to each vertex of an $n$-vertex graph $G$, then an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/r\rceil$…

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 consider the problem of coloring k-colorable graphs with the fewest possible colors. We present a randomized polynomial time algorithm that colors a 3-colorable graph on $n$ vertices with min O(Delta^{1/3} log^{1/2} Delta log n),…

An equitable $k$-coloring of a graph is a proper $k$-coloring where the sizes of any two different color classes differ by at most one. In 1973, Meyer conjectured that every connected graph $G$ has an equitable $k$-coloring for some $k\leq…

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…

The topological Tverberg theorem has been generalized in several directions by setting extra restrictions on the Tverberg partitions. Restricted Tverberg partitions, defined by the idea that certain points cannot be in the same part, are…