Graphs and Combinatorics

, Volume 12, Issue 4, pp 385–395

# Hamiltonian cycles in 1-tough graphs

• Bing Wei
Article

## Abstract

For a graph G, let σ 3 = min{ i=1 3 d(u i ): {u 1, u 2, u 3} is an independent set of G} and $$\bar \sigma _3 = \min \left\{ {\sum\nolimits_{i = 1}^3 {d\left( {u_i } \right) - \left| {\bigcap\nolimits_{i = 1}^3 {N\left( {u_i } \right)} } \right|:} } \right.\left\{ {u_1 ,u_2 ,u_3 } \right\}$$ is an independent set of G}. In this paper, we shall prove the following result: Let G be a 1-tough graph with n vertices such that σ 3 ≥ n and $$\bar \sigma _3 \geqslant n - 4$$. ThenG is hamiltonian. This generalizes a result of Fassbender [2], a result of Flandrin, Jung and Li [3] and a result of Jung [5].

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