Covering cycles andk-term degree sums

Abstract

We show that if\(\sum\limits_{x \in S} {\deg _{G^x } \geqslant \left| G \right|}\), for every stable set\(S \subseteq V\left( G \right),\left| S \right| = k\), then the vertex set ofG can be covered withk−1 cycles, edges or vertices. This settles a conjecture by Enomoto, Kaneko and Tuza.

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References

  1. [1]

    G. A. Dirac: Some theorems on abstract graphs,Proc. London Math. Soc.,2 (1952), 69–81.

    Google Scholar 

  2. [2]

    H. Enomoto, A. Kaneko, M. Kouider andZs. Tuza: Degree sums and covering cycles,Journal of Graph Theory,20 (1995), 419–422.

    Google Scholar 

  3. [3]

    H. Enomoto, A. Kaneko, andZs. Tuza:P 3-factors and covering cycles in graphs of minimum degreen/3, Colloquia Mathematica Societatis János Bolyai 52. Combinatorics, Eger (Hungary), North Holland (1988), 213–220.

    Google Scholar 

  4. [4]

    M. Kouider: Covering vertices by cycles,Journal of Graph Theory,18 (1994), 757–776.

    Google Scholar 

  5. [5]

    O. Ore: Note on Hamiltonian circuits,Amer. Math. Monthly,67 (1960), 55.

    Google Scholar 

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Kouider, M., Lonc, Z. Covering cycles andk-term degree sums. Combinatorica 16, 407–412 (1996). https://doi.org/10.1007/BF01261324

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Mathematics Subject Classification (1991)

  • 05 C 38
  • 05 C 70