Maximum sparse induced subgraphs of the binomial random graph with given number of edges

TL;DR

This paper investigates the maximum size of induced subgraphs with a given number of edges in the binomial random graph, showing concentration results for these sizes in probabilistic settings.

Contribution

It establishes concentration results for the maximum size of induced subgraphs with a specified number of edges in G(n,p), extending understanding of subgraph structures.

Findings

Maximum size of induced subtrees is concentrated in 2 points.

Maximum size of induced subgraphs with a given number of edges is concentrated in 2 points.

Results hold under certain smoothness conditions on the edge count function.

Abstract

We prove that a.a.s. the maximum size of an induced subtree of the binomial random graph $G(n,p)$ is concentrated in 2 consecutive points. We also prove that, given a non-negative integer-valued function $t(k)<\varepsilon k^2$, under a certain smoothness condition on this function, a.a.s. the maximum size $k$ of an induced subgraph with exactly $t(k)$ edges of $G(n,p)$ is concentrated in 2 consecutive points as well.