Connectivity keeping trees in triangle-free graphs

TL;DR

This work extends connectivity-preserving subgraph results to triangle-free graphs by showing that for any tree $T$ of order $m$, every $k$-connected triangle-free graph $G$ with minimum degree $\delta(G)\ge 2k+3m-4$ contains a subtree $T'\cong T$ such that $\kappa(G-V(T'))\ge k$. The authors harness the notion of a $p$-connected triple and a saturating matching to locate a large induced subgraph where $T$ can be embedded via standard tree-embedding results (and their bipartite/girth-restricted variants). The triangle-free condition is crucial to bound neighborhood overlaps and guarantee the necessary matchings, enabling the preservation of $k$-connectivity after removing $V(T')$. They also provide refined results for special graph classes, including bipartite graphs and graphs with girth at least five, thereby advancing the broader program of Mader-type connectivity keeping. The work connects to Erdős–Sós-type tree embedding questions and suggests avenues for further improvements and generalizations beyond triangle-free graphs.

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.

Figures (2)

• Figure 1: A $p$-connected triple $(S_1,S_2,F)$ of $G$, where the region surrounded by dotted lines is related to the condition (iii) of Definition \ref{['def:contri']}
• Figure 2: An illustration for the proof of Lemma \ref{['lem:k-connected']}

Theorems & Definitions (22)

• Theorem 1.1: CKL72
• Theorem 1.2: Mader10
• Conjecture 1.3: Mader10
• Theorem 1.4: LYH24
• Theorem 1.5: LTW22
• Conjecture 1.6: LTW22
• Theorem 1.7
• Theorem 1.8
• Lemma 2.1: Chvatal77
• Lemma 2.2: HWL25