Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Chvátal-Erdős condition for 2-factors with at most two components in graphs

Tao Tian, Liming Xiong, Weigen Yan

Abstract

It is well-known that Chvátal and Erdős 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.

Chvátal-Erdős condition for 2-factors with at most two components in graphs

Abstract

It is well-known that Chvátal and Erdős 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.
Paper Structure (4 sections, 8 theorems, 3 equations, 4 figures)

This paper contains 4 sections, 8 theorems, 3 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Th1']}
  3. Sharpness of Theorem \ref{['Th1']}
  4. Concluding remark

Key Result

Theorem 1.1

(Chvátal and Erdős 3) Let $G$ be a connected graph of order at least $3$. Then each of the following holds. (a) If $\alpha(G)\leq \kappa(G)$, then $G$ is Hamiltonian. (b) If $\alpha(G)\leq \kappa(G)+1$, then $G$ is traceable. (c) If $\alpha(G)\leq \kappa(G)-1$, then $G$ is Hamilton-connected.

Figures (4)

  • Figure 1: The graphs $G_1$,$G_2$ and $G_3$.
  • Figure 2: (a) The case when $u_{2}^{+}u_{1}^{++}\in E(G)$. (b) The case when $u_{1}^{++}u_{l}^{-}\in E(G)$.
  • Figure 3: (a) The case when $u_{1}^{+}u_{j}^{++}\in E(G)$. (b) The case when $j=i$.
  • Figure 4: The graph $G=(k+1)K_{2}\vee kK_{1}$.

Theorems & Definitions (21)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Claim 1
  • Claim 2
  • ...and 11 more