Chv\'{a}tal-Erd\H{o}s condition for 2-factors with at most two components in graphs

TL;DR

This paper extends classical graph theory conditions, showing that under certain independence and connectivity constraints, graphs have 2-factors with at most two components, with the result being optimal.

Contribution

It establishes a new sufficient condition for the existence of 2-factors with limited components based on independence and connectivity, improving previous results.

Findings

Graphs with order at least three times their connectivity plus three have a 2-factor with at most two components.

The result applies when the independence number is no greater than connectivity plus one.

The theorem is proven to be best possible.

Abstract

It is well-known that Chv\'{a}tal and Erd\H{o}s stated that any graph of order at least three whose independence number is no greater than its connectivity is Hamiltonian; that any graph whose independence number is no greater than its connectivity minus one is Hamilton-connected; and that any graph whose independence number is no greater than its connectivity plus one is traceable. Kaneko and Yoshimoto [J. Graph Theory 43 (2003) 269--279] showed that every 4-connected graph of order at least six has a 2-factor with two components if its independence number is no greater than its connectivity. In this paper, we show that any connected graph of order at least three times its connectivity plus three has a 2-factor with at most two components, except for one exceptional class, if its independence number is no greater than its connectivity plus one. Our result is best possible.