Coloring non-uniform hypergraphs: a new algorithmic approach to the general Lovász local lemma

Equitable Coloring of k-Uniform Hypergraphs

Let H be a k-uniform hypergraph with n vertices. A strong r-coloring is a partition of the vertices into r parts, such that each edge of H intersects each part. A strong r-coloring is called equitable if the size of each part is dn/re or bn/rc. We prove that for all a ≥ 1, if the maximum degree of H satisfies ∆(H) ≤ k then H has an equitable coloring with k a ln k (1 − ok(1)) parts. In particul...

An alternative proof for the constructive Asymmetric Lovász Local Lemma

We provide an alternative constructive proof of the Asymmetric Lovász Local Lemma. Our proof uses the classic algorithmic framework of Moser and the analysis introduced by Giotis, Kirousis, Psaromiligkos, and Thilikos in " On the algorithmic Lovász Local Lemma and acyclic edge coloring " , combined with the work of Bender and Richmond on the multivariable Lagrange Inversion formula.

A (1+epsilon)-approximation algorithm for partitioning hypergraphs using a new algorithmic version of the Lovász Local Lemma

A note on near-optimal coloring of shift hypergraphs

As shown in the original work on the Lovász Local Lemma due to Erdős & Lovász (Infinite and Finite Sets, 1975), a basic application of the Local Lemma answers an infinitary coloring question of Strauss, showing that given any integer set S, the integers may be k-colored so that S and all its translates meet every color. The quantitative bounds here were improved by Alon, Kriz & Nes̆etr̆il (Studia...

Greedy colorings of uniform hypergraphs

We give a very short proof of an Erdős conjecture that the number of edges in a non-2-colorable n-uniform hypergraph is at least f(n)2, where f(n) goes to infinity. Originally it was solved by József Beck in 1977, showing that f(n) at least c log n. With an ingenious recoloring idea he later proved that f(n) ≥ cn. Here we prove a weaker bound on f(n), namely f(n) ≥ cn. Instead of recoloring a r...