A sufficient condition for pancyclability of graphs

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Abstract

Let G be a graph of order n and S be a vertex set of q vertices. We call G, S-pancyclable, if for every integer i with 3 ≤ i ≤ q there exists a cycle C in G such that | V (C) ∩ S | = i. For any two nonadjacent vertices u, v of S, we say that u, v are of distance two in S, denoted by dS (u, v) = 2, if there is a path P in G connecting u and v such that | V (P) ∩ S | ≤ 3. In this paper, we will prove that if G is 2-connected and for all pairs of vertices u, v of S with dS (u, v) = 2, max {d (u), d (v)} ≥ frac(n, 2), then there is a cycle in G containing all the vertices of S. Furthermore, if for all pairs of vertices u, v of S with dS (u, v) = 2, max {d (u), d (v)} ≥ frac(n + 1, 2), then G is S-pancyclable unless the subgraph induced by S is in a class of special graphs. This generalizes a result of Fan [G. Fan, New sufficient conditions for cycles in graphs, J. Combin. Theory B 37 (1984) 221-227] for the case when S = V (G). © 2008 Elsevier B.V. All rights reserved.

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Flandrin, E., Li, H., & Wei, B. (2009). A sufficient condition for pancyclability of graphs. Discrete Mathematics, 309(1), 144–150. https://doi.org/10.1016/j.disc.2007.12.063

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