A generalization of the Chvátal-Erdős theorem

TL;DR

The paper generalizes the Chvátal–Erdős theorem by replacing the independence-number condition with the broader $[s,t]$-graph framework. It proves that every $k$-connected $[k+1,2]$-graph is Hamiltonian-connected except the extremal family $G = kK_1 \vee G_k$ with $|G_k|=k$, thus extending CE. The proof employs a longest $(u,v)$-path argument and a series of structural claims on the neighborhoods of the path to show that, unless the extremal join structure occurs, Hamiltonian-connectedness holds. This work connects independence-like conditions to induced-subgraph edge-density, clarifying when Hamiltonian-connectivity arises under generalized local sparsity, and it includes an accompanying extremal-bound discussion via a double-counting argument.

Abstract

A well-known result of Chvátal and Erdős from 1972 states that a graph with connectivity not less than its independence number plus one is hamiltonian-connected. A graph $G$ is called an $[s,t]$-graph if any induced subgraph of $G$ of order $s$ has size at least $t.$ We prove that every $k$-connected $[k+1,2]$-graph is hamiltonian-connected except $kK_1\vee G_{k},$ where $k\ge 2$ and $G_{k}$ is an arbitrary graph of order $k.$ This generalizes the Chvátal-Erdős theorem.

Theorems & Definitions (18)

• Theorem 1.1: Chvátal-Erdős CE1
• Theorem 1.2
• Remark 1
• Claim 1
• Claim 2
• proof : Proof of Claim \ref{['c2']}
• Claim 3
• proof : Proof of Claim \ref{['c3']}
• Claim 4
• proof : Proof of Claim \ref{['c4']}