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The New Lower Bound of the Number of Vertices of Degree 5 in Contraction Critical 5-Connected Graphs

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Abstract

An edge of a k-connected graph is said to be k-contractible if its contraction results in a k-connected graph. A k-connected non-complete graph with no k-contractible edge, is called contraction critical k-connected. Let G be a contraction critical 5-connected graph, in this paper we show that G has at least \({\frac{1}{2}|G|}\) vertices of degree 5.

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References

  1. Ando K., Kawarabayashi K., Kaneko A.: Vertices of degree 5 in a contraction critically 5-connected graphs. Graphs Combin. 21, 27–37 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bondy J.A., Murty U.S.R.: Graph Theory with Application. Macmillan, New York (1976)

    Google Scholar 

  3. Egawa Y.: Contractible edges in n-connected graphs with minimum degree greater than or equal to \({[\frac{5n}{4}]}\). Graphs Combin. 7(1), 15–21 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. Kriesell M.: A degree sum condition for the existence of a contractible edge in a k-connected graph. J. Combin. Theory Ser. B 82, 81–101 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Mader W.: Generalizations of critical connectivity of graphs. Discrete Math. 72, 267–283 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  6. Martinov N.: Uncontractable 4-connected graphs. J. Graph Theory 6, 343–344 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  7. Qin C., Yuan X., Su J.: Some properties of contraction-critical 5-connected graphs. Discrete Math. 308, 5742–5756 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Su, J.: The vertices of degree 5 in contraction critical 5-connected graph. J. Guangxi Normal University 3, 12–16 (1997). (in Chinese)

    Google Scholar 

  9. Thomassen C.: Nonseparating cycles in k-connected graphs. J. Graph Theory 5, 351–354 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tutte W.T.: A theory of 3-connected graphs. Nederl. Akad. Wet. Proc. Ser. A 64, 441–455 (1961)

    MATH  MathSciNet  Google Scholar 

  11. Yuan, X.: The contractible edges of 5-connected graphs. J. Guangxi Normal University 3, 30–32 (1994). (in Chinese)

    Google Scholar 

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Authors and Affiliations

  1. Department of Public Teaching, Guangxi Economic Management Cadre College, Nanning, 530007, People’s Republic of China

    Tingting Li

  2. Department of Mathematics, Guangxi Normal University, Guilin, 541004, People’s Republic of China

    Jianji Su

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  1. Tingting Li
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  2. Jianji Su
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Correspondence to Tingting Li.

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Li, T., Su, J. The New Lower Bound of the Number of Vertices of Degree 5 in Contraction Critical 5-Connected Graphs. Graphs and Combinatorics 26, 395–406 (2010). https://doi.org/10.1007/s00373-010-0907-z

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  • DOI: https://doi.org/10.1007/s00373-010-0907-z

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