On degree sum conditions for long cycles and cycles through specified vertices

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Abstract

Let G be a graph. For S ⊂ V (G), let Δk (S) denote the maximum value of the degree sums of the subsets of S of order k. In this paper, we prove the following two results. (1) Let G be a 2-connected graph. If Δ2 (S) ≥ d for every independent set S of order κ (G) + 1, then G has a cycle of length at least min {d, | V (G) |}. (2) Let G be a 2-connected graph and X a subset of V (G). If Δ2 (S) ≥ | V (G) | for every independent set S of order κ (X) + 1 in G [X], then G has a cycle that includes every vertex of X. This suggests that the degree sum of nonadjacent two vertices is important for guaranteeing the existence of these cycles. © 2007 Elsevier B.V. All rights reserved.

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Yamashita, T. (2008). On degree sum conditions for long cycles and cycles through specified vertices. Discrete Mathematics, 308(24), 6584–6587. https://doi.org/10.1016/j.disc.2007.10.048

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