Connectivity keeping trees in 3-connected bipartite graphs with girth conditions

Abstract

Luo, Tian and Wu conjectured in 2022 that for any tree $T$ with bipartition $X$ and $Y$, every $k$-connected bipartite graph $G$ with $\delta (G)\ge k+t$, where $t=\max \{|X|,|Y|\}$, contains a subtree $T^{\prime }\cong T$ such that $G-V(T^{\prime })$ remains $k$-connected. This conjecture has been proved for caterpillars and spiders when $k\le 3$; and for paths with odd order. In this paper, we prove that this conjecture holds if $G$ is a bipartite graph with $g(G)\ge diam(T)-1$ and $k\le 3$, where $g(G)$ and $diam(T)$ denote the girth of $G$ and the diameter of $T$, respectively.

There was an error in the proof of the mian result Theorem 3.1