On the independence number of random trees via tricolourations

TL;DR

This paper investigates the independence number and related parameters of large random trees using a novel tricolouration method, deriving limit theorems for these quantities in various classes of simply generated trees.

Contribution

It introduces a canonical tricolouration approach to analyze independence and related parameters in large random trees, providing new limit theorems.

Findings

Limit theorems in $L^p$ for the renormalized independence number

Application to size-conditioned Bienaymé-Galton-Watson trees

Unified framework for independence, matching, and kernel dimension

Abstract

We are interested in the independence number of large random simply generated trees and related parameters, such as their matching number or the kernel dimension of their adjacency matrix. We express these quantities using a canonical tricolouration, which is a way to colour the vertices of a tree with three colours. As an application we obtain limit theorems in $L^p$ for the renormalised independence number in large simply generated trees (including large size-conditioned Bienaym\'e-Galton-Watson trees).