**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31533

##### On Chvátal’s Conjecture for the Hamiltonicity of 1-Tough Graphs and Their Complements

**Authors:**
Shin-Shin Kao,
Yuan-Kang Shih,
Hsun Su

**Abstract:**

In this paper, we show that the conjecture of Chv tal, which states that any 1-tough graph is either a Hamiltonian graph or its complement contains a specific graph denoted by F, does not hold in general. More precisely, it is true only for graphs with six or seven vertices, and is false for graphs with eight or more vertices. A theorem is derived as a correction for the conjecture.

**Keywords:**
Complement,
degree sum,
Hamiltonian,
tough.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1340542

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