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Minimum degree, edge-connectivity and radius

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Abstract

Let G be a connected graph on n≥4 vertices with minimum degree δ and radius r. Then \(\delta r\leq4\lfloor\frac{n}{2}\rfloor-4\), with equality if and only if one of the following holds:

  1. (1)

    G is K 5,

  2. (2)

    G≅K n ∖M, where M is a perfect matching, if n is even,

  3. (3)

    δ=n−3 and Δ≤n−2, if n is odd.

This solves a conjecture on the product of the edge-connectivity and radius of a graph, which was posed by Sedlar, Vukičević, Aouchice, and Hansen.

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References

  • Beezer RA, Riegsecker JE, Smith BA (2001) Using minimum degree to bound average distance. Discrete Math 226:365–371

    Article  MathSciNet  MATH  Google Scholar 

  • Czabarka É, Dankelmann P, Székely LA (2009) Diameter of 4-colourable graphs. Eur J Comb 30:1082–1089

    Article  MATH  Google Scholar 

  • Dankelmann P, Volkmann L (2009) Minimum size of a graph or digraph of given radius. Inf Process Lett 109:971–973

    Article  MathSciNet  MATH  Google Scholar 

  • Dankelmann P, Dlamini G, Swart HC (2005) Upper bounds on distance measures in K3,3-free graphs. Util Math 67:205–221

    MathSciNet  MATH  Google Scholar 

  • Dankelmann P, Mukwembi S, Swart HC (2007) An upper bond on the radius of 3-edge-connected graph. Util Math 73:207–215

    MathSciNet  MATH  Google Scholar 

  • Dlamini G (2003) Aspects of distances in graphs. Ph.D. thesis, University of Natal, Durban

  • Erdős P, Pach J, Pollack R, Tuza Z (1989) Radius, diameter, and minimum degree. J Comb Theory, Ser B 47:73–79

    Article  Google Scholar 

  • Fajtlowicz S (1988) A characterization of radius-critical graphs. J Graph Theory 12:529–532

    Article  MathSciNet  MATH  Google Scholar 

  • He W, Li S (1991) The average distance of chain regular graphs. Math Appl 4:28–34 (in Chinese)

    MATH  Google Scholar 

  • Henning MA, Mukwembi S (2009) Domination, radius, and minimum degree. Discrete Appl Math 157:2964–2968

    Article  MathSciNet  MATH  Google Scholar 

  • Jolivet JL (1972) Sur la connexité des graphes orientés. C R Math 274:148–150

    MathSciNet  MATH  Google Scholar 

  • Kouider M, Winkler P (1997) Mean distance and minimum degree. J Graph Theory 25:95–99

    Article  MathSciNet  MATH  Google Scholar 

  • Plesník J (1975) Critical graphs of given diameter. Acta Fac Rerum Nat Univ Commenian Math 30:71–93

    MATH  Google Scholar 

  • Sedlar J, Vukičević D, Aouchice M, Hansen P (2008) Variable neighborhood search for extremal graphs: 25. Products of connectivity and distance measures. Graph Theory Notes N Y LV:6–13

    Google Scholar 

  • Van den Berg P (2007) Bounds on distance parameters of graphs, Ph.D. thesis, University of KwaZulu-Natal

  • West DB (2001) Introduction to graph theory, 2nd edn. Prentice Hall, Upper Saddle River

    Google Scholar 

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Acknowledgements

The authors would like to thank the referees whose detailed comments led to significant improvement in the clarity of our presentation. Research supported by the key project of Chinese Ministry of Education (No. 210243), NSFC (No. 11161046), Science Fund for Creative Research Groups (No. 11021161), XJEDU2009S20. The research of the fifth author is supported by the research grant of Xinjiang Polytechnical University.

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Authors and Affiliations

  1. College of Mathematics and System Science, Xinjiang University, Urumqi, Xinjiang, 830046, P.R. China

    Baoyindureng Wu, Xinhui An & Guojie Liu

  2. Academy of Mathematics and System Science, Chinese Academy of Sciences, Beijing, 100080, P.R. China

    Guiying Yan

  3. Xinjiang Polytechnical College, Urumqi, Xinjiang, 830000, P.R. China

    Xiaoping Liu

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  1. Baoyindureng Wu
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  2. Xinhui An
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  3. Guojie Liu
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  4. Guiying Yan
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  5. Xiaoping Liu
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Correspondence to Baoyindureng Wu.

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Wu, B., An, X., Liu, G. et al. Minimum degree, edge-connectivity and radius. J Comb Optim 26, 585–591 (2013). https://doi.org/10.1007/s10878-012-9479-6

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  • DOI: https://doi.org/10.1007/s10878-012-9479-6

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