The optimal edge-colouring threshold

TL;DR

This paper determines the threshold probability for properly edge-colouring dense bipartite graphs, Steiner Triple Systems, and Latin squares, resolving several open questions in combinatorics.

Contribution

It establishes the exact order of the threshold probability for L-edge-colourability in dense quasirandom bipartite graphs, Steiner Triple Systems, and Latin squares.

Findings

Threshold probability is of order (log n)/n for proper L-edge-colouring.

Answers the question of Kang et al. on bipartite graphs.

Provides the same threshold for Latin squares, answering Johanssen's 2006 question.

Abstract

Consider any dense r-regular quasirandom bipartite graph H with parts of size n and fix a set of r colours. Let L be a random list assignment where each colour is available for each edge of H with probability p. We show that the threshold probability for H to have a proper L-edge-colouring is p of order (log n)/n. This answers a question of Kang, Kelly, K\"uhn, Methuku and Osthus. We thus obtain the same threshold for Steiner Triple Systems and Latin squares; the latter answers a question of Johanssen from 2006.