Given a graph H, a graph G is H-free if G does not contain H as an induced subgraph. For a positive real number t, a non-complete graph G is said to be t-tough if for every vertex cut S of G, the ratio of |S| to the number of components of G − S is at least t. A complete graph is said to be t-tough for any t > 0. Chvátal's toughness conjecture, stating that there exists a constant t 0 such that every t 0 -tough graph with at least three vertices is Hamiltonian, is still open in general. Chvátal and Erdös [8] proved that, for any integer k ≥ 1, every max{2, k}-connected (k + 1)P 1 -free graph on at least three vertices is Hamiltonian. Along the Chvátal-Erdös theorem, Shi and Shan [18] proved that, for any integer k ≥ 4, every 4-tough 2kconnected (P 2 ∪kP 1 )-free graph with at least three vertices is Hamiltonian, and furthermore, they proposed a conjecture that for any integer k ≥ 1, any 1-tough 2k-connected (P 2 ∪kP 1 )-free graph is Hamiltonian. In this paper, we confirm the conjecture, and furthermore, we show that if k ≥ 3, then the condition '2k-connected' may be weakened to be '2(k − 1)-connected'. As an immediate consequence, for any integer k ≥ 3, every (k − 1)-tough (P 2 ∪ kP 1 )-free graph is Hamiltonian. This improves the result of Hatfield and Grimm [11], stating that every 3-tough (P 2 ∪ 3P 1 )-free graph is Hamiltonian.
In this paper, we give tight bounds for the normalized Laplacian eigenvalues of hypergraphs that are not necessarily uniform, and provide edge version interlacing inequalities, Cheeger inequalities, and a discrepancy inequality that are related to the normalized Laplacian eigenvalues for uniform hypergraphs.
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