Sufficient average degree conditions for the existence of large highly connected subgraphs

TL;DR

This paper improves the known average degree thresholds for guaranteeing large, highly connected subgraphs in graphs, advancing from previous bounds to a new threshold of 3.109, with larger subgraphs also guaranteed.

Contribution

It establishes a new average degree condition of 3.109 for the existence of large, highly connected subgraphs, refining prior bounds and including size constraints.

Findings

Improved the average degree threshold to 3.109 for large highly connected subgraphs.

Proved that larger $(k+1)$-connected subgraphs exist under similar average degree conditions.

Demonstrated that the subgraphs can have more than 1.2k vertices.

Abstract

Mader proved that every sufficiently large graph with average degree at least $(2+\sqrt{2})k$ has a $(k+1)$-connected subgraph. He also conjectured that an average degree of at least $3k$ is sufficient. The best known sufficient factor was improved by multiple authors but never reached $3$. In the present paper, it is further improved to $3.109$. In addition, the obtained $(k+1)$-connected subgraph is constrained to have more than $1.2k$ vertices. Moreover, similar conditions on the average degree are proven to be sufficient for the existence of even greater $(k+1)$-connected subgraphs.