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**A result on Hamiltonian line graphs involving restrictions on induced subgraphs.**
*(English)*
Zbl 0651.05049

Author’s abstract: “It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on Hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a Hamiltonian line graph.”

As a consequence of the main result in this paper it is deduced that L(G) is Hamiltonian for every graph G of order \(n\geq 4\) such that \(d(u)+d(v)\geq n-1\) holds either for every pair u, v of nonadjacent vertices, either for every edge uv\(\in E(G)\neq \emptyset\) whenever \(G\not\cong P_ 4\).

As a consequence of the main result in this paper it is deduced that L(G) is Hamiltonian for every graph G of order \(n\geq 4\) such that \(d(u)+d(v)\geq n-1\) holds either for every pair u, v of nonadjacent vertices, either for every edge uv\(\in E(G)\neq \emptyset\) whenever \(G\not\cong P_ 4\).

Reviewer: I.Tomescu

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\textit{H. J. Veldman}, J. Graph Theory 12, No. 3, 413--420 (1988; Zbl 0651.05049)

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