Highly connected triples and Mader's conjecture

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