Work

A random forest process with a variable number of giant components in the threshold window

Year: 2025

Type: article

Abstract: Given a graph G , and an ordering π of its vertices, a permutation forest F ( G , π ) is a spanning forest of G whose components are obtained as follows. For each vertex v , connect v to its first neighbour w in G that appears after v in the ordering π . If we regard this edge ( v , w ) as directed forward, from v to w , then each vertex has at most one forward edge, and the components of F ( G , π ) are arborescences. The roots of the components formed by this process are those vertices of G with no forward edge in the ordering π . This paper shows that the permutation forests of the random graphs G n , p have a threshold for the emergence of a linear size component around p = 1 / n . In contrast to the w.h.p. emergence of a unique giant in G n , p , the permutation forest process has the unusual property that, with positive probability, a number of linear size components occur within the threshold window.

Source: Discrete Applied Mathematics

DOI: 10.1016/j.dam.2025.12.030

Authors: Colin Cooper, Tomasz Radzik

Institution: King's College London

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: $3,090

Funders: King's College London, Engineering and Physical Sciences Research Council