Work

The emergence of a giant rainbow component

Year: 2025

Type: article

Abstract: The random coloured graph G c ( n , p ) is obtained from the Erdős–Rényi binomial random graph G ( n , p ) by assigning to each edge a colour from a set of c colours independently and uniformly at random. It is not hard to see that, when c = Θ ( n ) , the order of the largest rainbow tree in this model undergoes a phase transition at the critical point p = 1 n . In this paper, we determine the asymptotic order of the largest rainbow tree in the weakly sub- and supercritical regimes , when ɛ p = 1 + ɛ n for some ɛ ɛ ɛ = ɛ ( n ) which satisfies ɛ ɛ = o ( 1 ) and ɛ | ɛ | 3 n → ∞ . In particular, we show that in both of these regimes with high probability the largest component of G c ( n , p ) contains a rainbow tree which is almost spanning. We also consider the order of the largest rainbow tree in the sparse regime , when p = d n for some constant d > 1 . Here we show that the largest rainbow tree has linear order, and, moreover, for d and c sufficiently large, with high probability G c ( n , p ) even contains a rainbow cycle which is almost spanning.

Source: European Journal of Combinatorics

DOI: 10.1016/j.ejc.2025.104154

Authors: Oliver Cooley, Tuan Anh Do, Joshua Erde, Michael Missethan

Institutions: Ludwig-Maximilians-Universität München, Beijing Institute of Technology, Graz University of Technology

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,560

Funder: Austrian Science Fund