Abstract The Erdős--Faber--Lovász conjecture (posed in 1972) states that the chromatic
index of any linear hypergraph on n vertices is at most n. In this paper, we prove this …

arxiv:2210.05915v2 [math.CO] 22 Apr 2023 Page 1 arxiv:2210.05915v2 [math.CO] 22 Apr
2023 Coloring, List Coloring, and Painting Squares of Graphs (and other related problems) …

This paper provides a survey of methods, results, and open problems on graph and
hypergraph colourings, with a particular emphasis on semi-random “nibble” methods. We …

A PROOF OF THE ERDOS–FABER–LOVASZ CONJECTURE 1. Introduction Graph and
hypergraph colouring problems are central to comb Page 1 A PROOF OF THE ERDOS–FABER–LOVASZ …

Following the groundbreaking algorithm of Moser and Tardos for the Lovasz Local Lemma
(LLL), there has been a plethora of results analyzing local search algorithms for various …

An independent set may not contain both a vertex and one of its neighbours. This basic fact
makes the uniform distribution over independent sets rather special. We consider the hard …

By a theorem of Johansson, every triangle-free graph G of maximum degree Δ has
chromatic number at most (C+ o (1)) Δ/log⁡ Δ for some universal constant C> 0. Using the …

A graph $ G $ is $ k $-locally sparse if for each vertex $ v\in V (G) $, the subgraph induced
by its neighborhood contains at most $ k $ edges. Alon, Krivelevich, and Sudakov showed …

We analyse uniformly random proper k-colourings of sparse graphs with maximum degree Δ
in the regime Δ< k ln k. This regime corresponds to the lower side of the shattering threshold …

A conjecture of Alon, Krivelevich and Sudakov states that, for any graph, there is a constant
such that if is an-free graph of maximum degree, then. Alon, Krivelevich and Sudakov …