Induced forests in regular graphs with large girth

TL;DR

This paper analyzes a randomized algorithm for growing induced forests in regular graphs with large girth, providing bounds that match known asymptotic results for random regular graphs, and offers an alternative proof for the random case.

Contribution

It introduces a simple randomized algorithm for induced forests in regular graphs with large girth and establishes bounds that align with known asymptotic results, offering a new proof for the random case.

Findings

Expected size of induced forest matches known bounds for large girth graphs.

Provides an alternative proof for bounds in random regular graphs.

Algorithm performs well in graphs with degree at least 4 and large girth.

Abstract

An induced forest of a graph G is an acyclic induced subgraph of G. The present paper is devoted to the analysis of a simple randomised algorithm that grows an induced forest in a regular graph. The expected size of the forest it outputs provides a lower bound on the maximum number of vertices in an induced forest of G. When the girth is large and the degree is at least 4, our bound coincides with the best bound known to hold asymptotically almost surely for random regular graphs. This results in an alternative proof for the random case.