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On a Sharp Degree Sum Condition for Disjoint Chorded Cycles in Graphs

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Abstract

Let r and s be nonnegative integers, and let G be a graph of order at least 3r + 4s. In Bialostocki et al. (Discrete Math 308:5886–5890, 2008), conjectured that if the minimum degree of G is at least 2r + 3s, then G contains a collection of r + s vertex-disjoint cycles such that s of them are chorded cycles, and they showed that the conjecture is true for r = 0, s = 2 and for s = 1. In this paper, we settle this conjecture completely by proving the following stronger statement; if the minimum degree sum of two nonadjacent vertices is at least 4r + 6s−1, then G contains a collection of r + s vertex-disjoint cycles such that s of them are chorded cycles.

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Authors and Affiliations

  1. Department of Mathematical Information Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjuku-ku, Tokyo, 162-8601, Japan

    Shuya Chiba

  2. Department of Mathematics, Gunma National College of Technology, 580 Toriba, Maebashi, Gunma, 371-8530, Japan

    Shinya Fujita

  3. School of Mathematics, Ningxia University, Yinchuan, 750021, People’s Republic of China

    Yunshu Gao

  4. School of Mathematics, Shandong University, Jinan, 250100, People’s Republic of China

    Guojun Li

Authors
  1. Shuya Chiba
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  2. Shinya Fujita
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  3. Yunshu Gao
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  4. Guojun Li
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Correspondence to Shuya Chiba.

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Chiba, S., Fujita, S., Gao, Y. et al. On a Sharp Degree Sum Condition for Disjoint Chorded Cycles in Graphs. Graphs and Combinatorics 26, 173–186 (2010). https://doi.org/10.1007/s00373-010-0901-5

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  • DOI: https://doi.org/10.1007/s00373-010-0901-5

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