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 $\delta(G) \geq k + t$, where $t = \max\{|X|,|Y |\}$, contains a subtree $T' \cong T$ such that $G-V(T')$ remains $k$-connected. This conjecture has been proved for caterpillars and spiders when $k\leq 3$; and for paths with odd order. In this paper, we prove that this conjecture holds if $G$ is a bipartite graph with $g(G)\geq diam(T)-1$ and $k\leq 3$, where $g(G)$ and $diam(T)$ denote the girth of $G$ and the diameter of $T$, respectively.