Highly connected triples and Mader's conjecture

Abstract

Mader proved that, for any tree $T$ of order $m$, every $k$-connected graph $G$ with $\delta \left(G\right)\ge 2{\left(k+m-1\right)}^2+m-1$ contains a subtree $T^{\prime }\cong T$ such that $G-V\left(T^{\prime }\right)$ is $k$-connected. We proved that any graph $G$ with minimum degree $\delta \left(G\right)\ge 2k$ contains $k$-connected triples. As a corollary, we prove that, for any tree $T$ of order $m$, every $k$-connected graph $G$ with $\delta \left(G\right)\ge 3k+4m-6$ contains a subtree $T^{\prime }\cong T$ such that $G-V\left(T^{\prime }\right)$ is still $k$-connected, improving Mader's condition to a linear bound.