Connectivity keeping trees in triangle-free graphs

TL;DR

This paper extends connectivity preservation results for trees in graphs by proving that in triangle-free graphs with high minimum degree, one can find a subtree isomorphic to any given tree while maintaining the graph's connectivity.

Contribution

It generalizes previous bipartite and bipartite-like results to triangle-free graphs, establishing new minimum degree conditions for connectivity-preserving tree embeddings.

Findings

In triangle-free graphs, a subtree isomorphic to any given tree can be embedded while preserving $k$-connectivity.

The minimum degree condition is at least $2k+3m-4$ for trees of order $m$ in $k$-connected triangle-free graphs.

Refined results are provided for bipartite graphs and graphs with girth at least five.

Abstract

In 2012, Mader conjectured that for any tree $T$ of order $m$, every $k$-connected graph $G$ with minimum degree at least $\lfloor \frac{3k}{2}\rfloor+m-1$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In 2022, Luo, Tian, and Wu considered an analogous problem for bipartite graphs and conjectured that for any tree $T$ with bipartition $(X,Y)$, every $k$-connected bipartite graph $G$ with minimum degree at least $k+\max\{|X|,|Y|\}$ contains a subtree $T'\cong T$ such that $G-V(T')$ remains $k$-connected. In this paper, we relax the bipartite assumption by considering triangle-free graphs and prove that for any tree $T$ of order $m$, every $k$-connected triangle-free graph $G$ with minimum degree at least $2k+3m-4$ contains a subtree $T' \cong T$ such that $G-V(T')$ remains $k$-connected. Furthermore, we establish refined results for specific subclasses such as bipartite graphs or graphs with girth at least five.