Long induced paths in a configuration model

TL;DR

This paper refines the understanding of the length of induced paths in sparse random graphs by analyzing a specific algorithm and introducing a new exploration method, providing precise asymptotic results.

Contribution

It offers a sharp asymptotic analysis of induced paths in a configuration model using a novel algorithm and explores extensions combining depth-first and breadth-first searches.

Findings

Sharp asymptotic for induced path length

Analysis of a new exploration algorithm

Extension to m-induced paths

Abstract

In an article published in 1987 in Combinatorica \cite{MR918397}, Frieze and Jackson established a lower bound on the length of the longest induced path (and cycle) in a sparse random graph. Their bound is obtained through a rough analysis of a greedy algorithm. In the present work, we provide a sharp asymptotic for the length of the induced path constructed by their algorithm. To this end, we introduce an alternative algorithm that builds the same induced path and whose analysis falls into the framework of a previous work by the authors on depth-first exploration of a configuration model \cite{EFMN}. We also analyze an extension of our algorithm that mixes depth-first and breadth-first explorations and generates $m$-induced paths.