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A note on a new result related to Chartrand, Kaigars and Lick's theorem

Zhong Huang, Meng Ji

TL;DR

The paper strengthens the Chartrand–Kaigars–Lick result by proving that in a $k$-connected graph with minimum degree $\delta(G)\ge \lfloor\frac{3k}{2}\rfloor$, any end associated with a minimum separating set is disjoint from all other minimum separating sets, and removal of a vertex from that end preserves connectivity $\kappa(G)$. The proof uses a detailed end/fragment decomposition to derive contradictions unless $F$ avoids all minimum separators, yielding a robust structure that immediately implies CKL as a corollary. As applications, the authors provide a simpler proof of Mader's path theorem, and discuss implications for related conjectures, including Fujita–Kawarabayashi's and Mader's conjectures on connectivity-preserving subgraphs. Overall, the work introduces a new method to construct connectivity-preserving subgraphs via end-structure, with clear implications for embedding trees and sustaining $k$-connectivity after deletions.

Abstract

In this note, we prove a theorem covering Chartrand, Kaigars, and Lick's theorem in [Proc. Amer. Math. Soc. 32 (1972), 63-68]. As an application, we give a simpler proof of theorem proved by Mader [J. Graph Theory 65 (2010), 61-69. (Theorem 1)].

A note on a new result related to Chartrand, Kaigars and Lick's theorem

TL;DR

The paper strengthens the Chartrand–Kaigars–Lick result by proving that in a -connected graph with minimum degree , any end associated with a minimum separating set is disjoint from all other minimum separating sets, and removal of a vertex from that end preserves connectivity . The proof uses a detailed end/fragment decomposition to derive contradictions unless avoids all minimum separators, yielding a robust structure that immediately implies CKL as a corollary. As applications, the authors provide a simpler proof of Mader's path theorem, and discuss implications for related conjectures, including Fujita–Kawarabayashi's and Mader's conjectures on connectivity-preserving subgraphs. Overall, the work introduces a new method to construct connectivity-preserving subgraphs via end-structure, with clear implications for embedding trees and sustaining -connectivity after deletions.

Abstract

In this note, we prove a theorem covering Chartrand, Kaigars, and Lick's theorem in [Proc. Amer. Math. Soc. 32 (1972), 63-68]. As an application, we give a simpler proof of theorem proved by Mader [J. Graph Theory 65 (2010), 61-69. (Theorem 1)].
Paper Structure (4 sections, 4 theorems)

This paper contains 4 sections, 4 theorems.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['them1']}
  3. Simpler proof of Theorem \ref{['made']}
  4. Concluding remarks

Key Result

Theorem 1

Every $k$-connected graph $G$ with $\delta (G)\geq \lfloor\frac{3k}{2}\rfloor$ has a vertex $x$ with $\kappa(G-x)\geq k$.

Theorems & Definitions (8)

  • Theorem 1: Chartrand, Kaigars and Lick CKL
  • Theorem 2
  • Theorem 3: Fujita and Kawarabayashi SK
  • Conjecture 1: Fujita and Kawarabayashi SK
  • Theorem 4: Mader mader1
  • Claim 1
  • proof
  • Conjecture 2: Mader mader1