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⁡{|X|,|Y|}, then G has a subtree TG isomorphic to T such that G−V(TG) is k-connected. Although this conjecture has been validated for spiders and caterpillars in cases where k≤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≤3 with the girth under of G at least the diameter of G minus one.

中文翻译：

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探索二分图中的冗余树


Luo 等人推测，对于具有二分 X 和 Y 的树 T，如果一个 k 连通的二分图 G 的最小度至少为 k+max{|X|，|Y|}，则 G 具有与 T 同构的子树 TG，因此 G−V（TG） 是 k 连通的。尽管这个猜想已经在 k≤3 的情况下对蜘蛛和毛毛虫以及奇数顺序的路径进行了验证，但其普遍适用性仍然是一个悬而未决的问题。在本文中，我们确定了 k≤3 的猜想的有效性，其中 G 的周长至少为 G 的直径减 1。