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Length of Longest Cycles in a Graph Whose Relative Length is at Least Two

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Abstract

Let G be a graph. We denote p(G) and c(G) the order of a longest path and the order of a longest cycle of G, respectively. Let κ(G) be the connectivity of G, and let σ 3(G) be the minimum degree sum of an independent set of three vertices in G. In this paper, we prove that if G is a 2-connected graph with p(G) − c(G) ≥ 2, then either (i) c(G) ≥ σ 3(G) − 3 or (ii) κ(G) = 2 and p(G) ≥ σ 3(G) − 1. This result implies several known results as corollaries and gives a new lower bound of the circumference.

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Authors and Affiliations

  1. National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Kenta Ozeki

  2. Department of Mathematics, Kinki University, Kowakae 3-4-1, Higashiosaka, Osaka, 577-8502, Japan

    Tomoki Yamashita

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  1. Kenta Ozeki
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  2. Tomoki Yamashita
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Correspondence to Kenta Ozeki.

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Ozeki, K., Yamashita, T. Length of Longest Cycles in a Graph Whose Relative Length is at Least Two. Graphs and Combinatorics 28, 859–868 (2012). https://doi.org/10.1007/s00373-011-1078-2

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  • DOI: https://doi.org/10.1007/s00373-011-1078-2

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