Exploring redundant trees in bipartite graphs

Abstract

Luo et al. conjectured that for a tree T with bipartition X and Y, if a k-connected bipartite graph G with minimum degree at least $k+\max \left\{\right|X|,|Y\left|\right\}$, then G has a subtree $T_G$ isomorphic to T such that $G-V\left(T_G\right)$ is k-connected. Although this conjecture has been validated for spiders and caterpillars in cases where $k\le 3$, and also for paths with odd order, its general applicability has remained an open question. In this paper, we establish the validity of this conjecture for $k\le 3$ with the girth under of G at least the diameter of G minus one.