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New Results on EX Graphs

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Abstract

By the extremal number ex(n; t) = ex(n; {C 3, C 4, . . . , C t }) we denote the maximum size (that is, number of edges) in a graph of order n > t and girth at least g ≥ t + 1. The set of all the graphs of order n, containing no cycles of length ≥ t, and of size ex(n; t), is denoted by EX(n; t) = EX(n; {C 3, C 4, . . . , C t }), these graphs are called EX graphs. In 1975, Erdős proposed the problem of determining the extremal numbers ex(n; 4) of a graph of order n and girth at least 5. In this paper, we consider a generalized version of this problem, for t ≥ 5. In particular, we prove that ex(29; 6) = 45, also we improve some lower bounds and upper bounds of ex u (n; t), for some particular values of n and t.

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References

  1. Alon N., Hoory S., Linial N.: The Moore bound for irregular graphs. Graphs Comb. 18, 53–57 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bollobás, B.: Extremal Graph Theory. Academic Press Inc., London 18 (1978)

  3. Chartrand G., Lesniak L.: Graphs and digraphs, 3rd edn. Chapman and Hall, London (1996)

    MATH  Google Scholar 

  4. Erdős P.: Some recent progress on extremal problems in graph theory. Congr. Numer. 14, 3–14 (1975)

    Google Scholar 

  5. Garnick D.K., Nieuwejaar N.A.: Non-isomorphic extremal graphs with three-cylces or four-cycles. J. Comb. Math. Combin. Comput. 12, 33–56 (1993)

    MathSciNet  Google Scholar 

  6. Garnick D.K., Kwong Y.H.H., Lazebnik F.: Extremal graphs without three-cycles or four-cycles. J. Graph Theory 17, 633–645 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lazebnik F., Ustimenko V.A., Woldar A.J.: A new series of dense graphs of high girth. Bull. Am. Math. Soc (N.S.) 32(1), 73–79 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  8. Simonovits, M.: Extremal Graph Theory, Selected Topics in Graph Theory 2. In: Beineke, L.W., Wilson, R.J. (eds.) Academic Press, London, pp. 161–200 (1983)

  9. Tang J., Lin Y., Balbuena C., Miller M.: Calculating the extremal number ex(n; {C 3, C 4, . . . , C t }). J. Discret. Appl. Math. 157(9), 2198–2206 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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  1. School of Electrical Engineering and Computer Science, The University of Newcastle, Newcastle, NSW, 2308, Australia

    Jianmin Tang, Yuqing Lin & Mirka Miller

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  1. Jianmin Tang
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  2. Yuqing Lin
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  3. Mirka Miller
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Correspondence to Jianmin Tang.

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Tang, J., Lin, Y. & Miller, M. New Results on EX Graphs. Math.Comput.Sci. 3, 119–126 (2010). https://doi.org/10.1007/s11786-009-0009-6

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  • DOI: https://doi.org/10.1007/s11786-009-0009-6

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