Discrete Mathematics (2008) 308(19) 4518-4529

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Let G = (V, E) be a connected graph of order n, t a real number with t ≥ 1 and M ⊆ V (G) with | M | ≥ frac(n, t) ≥ 2. In this paper, we study the problem of some long paths to maintain their one or two different endpoints in M. We obtain the following two results: (1) for any vertex v ∈ V (G), there exists a vertex u ∈ M and a path P with the two endpoints v and u to satisfy | V (P) | ≥ min { frac(4, 4 + t) dG (u) + frac(4 - 2 t, 4 + t), frac(2, 1 + t) dG (u) - 1, dG (u) + 1 - t }; (2) there exists either a cycle C to cover all vertices of M or a path P with two different endpoints u0 and up in M to satisfy | V (P) | ≥ min { n, frac(f (t), 1 + f (t)) (dG (u0) + dG (up)) - 2 t - frac(6, 1 + f (t)) }, where f (t) = min { frac(4, t), frac(2, t - 1) }. © 2007 Elsevier B.V. All rights reserved.

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Li, H., & Li, J. (2008). Long paths with endpoints in given vertex-subsets of graphs. *Discrete Mathematics*, *308*(19), 4518–4529. https://doi.org/10.1016/j.disc.2007.08.056

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