A generalization of the Chv\'{a}tal-Erd\H{o}s theorem

TL;DR

This paper generalizes the Chvátal-Erdős theorem by proving that certain highly connected [k+1,2]-graphs are hamiltonian-connected, with a specific exception involving disjoint unions, thus extending classical connectivity and Hamiltonian properties.

Contribution

It introduces a broader class of graphs called [s,t]-graphs and establishes their Hamiltonian connectivity under new conditions, extending a classical theorem.

Findings

Every k-connected [k+1,2]-graph is hamiltonian-connected except for a specific disjoint union case.

Generalizes the Chvátal-Erdős theorem to a wider class of graphs.

Identifies the precise exception case involving disjoint unions of complete graphs and arbitrary graphs.

Abstract

A well-known result of Chv\'{a}tal and Erd\H{o}s 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 Chv\'{a}tal-Erd\H{o}s theorem.