Isolating highly connected induced subgraphs

TL;DR

This paper proves that graphs with sufficiently high minimum degree contain highly connected induced subgraphs with controlled external neighborhood size, extending classical results and providing new bounds and variants.

Contribution

It generalizes Mader's classical result by establishing the existence of highly connected induced subgraphs with bounded external neighbors in graphs with high minimum degree.

Findings

Graphs with minimum degree > 2k^2-1 contain (k+1)-connected induced subgraphs with limited external neighbors.

New bounds and variants for the existence of highly connected induced subgraphs are provided.

Improved proof and bounds for the existence of high connectivity and chromatic number subgraphs in high chromatic graphs.

Abstract

We prove that any graph $G$ of minimum degree greater than $2k^2-1$ has a $(k+1)$-connected induced subgraph $H$ such that the number of vertices of $H$ that have neighbors outside of $H$ is at most $2k^2-1$. This generalizes a classical result of Mader, which states that a high minimum degree implies the existence of a highly connected subgraph. We give several variants of our result, and for each of these variants, we give asymptotics for the bounds. We also we compute optimal values for the case when $k=2$. Alon, Kleitman, Saks, Seymour, and Thomassen proved that in a graph of high chromatic number, there exists an induced subgraph of high connectivity and high chromatic number. We give a new proof of this theorem with a better bound.