Maximum induced trees and forests of bounded degree in random graphs

TL;DR

This paper investigates the maximum sizes of induced trees and forests with bounded degree in dense random graphs, establishing concentration results and filling gaps in the understanding of their asymptotic behavior.

Contribution

It proves new concentration results for maximum induced trees and forests with degree constraints in random graphs, using advanced probabilistic and analytical techniques.

Findings

Proves 2-point concentration for maximum sizes in dense graphs.

Shows concentration in explicit small intervals for sparse graphs.

Develops new enumeration results for labelled trees and forests.

Abstract

Asymptotic behaviour of maximum sizes of induced trees and forests has been studied extensively in last decades, though the overall picture is far from being complete. In this paper, we close several significant gaps: 1) We prove $2$-point concentration of the maximum sizes of an induced forest and an induced tree with maximum degree at most $\Delta$ in dense binomial random graphs $G(n,p)$ with constant probability $p$. 2) We show concentration in an explicit interval of size $o(1/p)$ for the maximum size of an induced forest with maximum degree at most $\Delta$ for $1/n\ll p=o(1)$. Our proofs rely on both the second moment approach, with the probabilistic part involving Talagrand's concentration inequality and the analytical part involving saddle-point analysis, and new results on enumeration of labelled trees and forests that might be of their own interest.