On the number of edges in a graph with no $(k+1)$-connected subgraphs

TL;DR

This paper advances bounds on the maximum number of edges in graphs without $(k+1)$-connected subgraphs, moving closer to Mader's conjecture by improving previous upper bounds.

Contribution

The authors improve the upper bound on edges in such graphs from 193/120 to 19/12 times k(n-k), for larger n relative to k.

Findings

Improved upper bound on edges to 19/12 k(n-k)

Progress towards Mader's conjecture for large n

Refined understanding of connectivity constraints in graphs

Abstract

Mader proved that for $k\geq 2$ and $n\geq 2k$, every $n$-vertex graph with no $(k+1)$-connected subgraphs has at most $(1+\frac{1}{\sqrt{2}})k(n-k)$ edges. He also conjectured that for $n$ large with respect to $k$, every such graph has at most $\frac{3}{2}\left(k - \frac{1}{3}\right)(n-k)$ edges. Yuster improved Mader's upper bound to $\frac{193}{120}k(n-k)$ for $n\geq\frac{9k}{4}$. In this note, we make the next step towards Mader's Conjecture: we improve Yuster's bound to $\frac{19}{12}k(n-k)$ for $n\geq\frac{5k}{2}$.