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Minimum order of loop networks of given degree and girth

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Abstract

Behzad, Chartrand and Wall proposed the conjecture that any regular digraph of degreer and girthg has ordern ≥ r(g − 1) + 1. The conjecture was proved in [3] for vertex transitive graphs. For Loop Networks the conjecture is equivalent to a theorem of Shepherdson in additive number theory. We show that, except for graphs of a particular structure, Loop Networks, and in general Abelian Cayley graphs, verify the stronger inequalityn ≥ (r + 1)(g − 1) − 1. This bound is best possible.

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References

  1. Cacceta, L., Häggkvist, R.: On minimal digraphs of given girth, Proc. of the Ninth Southeastern Conference on Combinatorics, Graph Theory and Computing, Utilitas Mathematica, Winnipeg 181–187 (1978)

    Google Scholar 

  2. Chvatál, V., Szemerédi, E.: Short Cycles in Directed Graphs. J. Comb. Theory, Ser. B35, 323–327 (1983)

    Google Scholar 

  3. Hamidoune, Y.O.: An application of connectivity theory in graphs to factorization of elements in groups. Europ. J. Comb.2, 349–355 (1981)

    Google Scholar 

  4. Hamidoune, Y.O.: On the connectivity of Cayley graphs. Europ. J. Comb.5, 309–312 (1984)

    Google Scholar 

  5. Hamidoune, Y.O.: Sur les atomes d'un graphe orienté. C.R. Acad. Sc. Paris A 1253–1256 (1977)

  6. Hamidoune, Y.O., Llado, A.S., Serra, O.: Vosperian and superconnected Abelian Cayley digraphs. Graphs and Combinatorics7, 143–152 (1991)

    Article  Google Scholar 

  7. Kemperman, J.H.B.: On small sums in abelian groups. Acta Math.103, 63–88 (1960)

    Google Scholar 

  8. Nishimura, T.: Short Cycles in Digraphs. Discrete Math.72, 295–298 (1988)

    Article  Google Scholar 

  9. Shepherdson, J.C.: On the addition of elements of a subsequence. J. Lond. Math. Soc.22, 85–88 (1947)

    Google Scholar 

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Authors and Affiliations

  1. Universite Pierre et Marie Curie, ER Combinatoire 17, 4 Place Jussieu, 75230, Paris, France

    Y. O. Hamidoune

  2. Universitat Politènica de Catalunya, Ap.30002, 08080, Barcelona, Spain

    A. S. Llado & O. Serra

Authors
  1. Y. O. Hamidoune
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  2. A. S. Llado
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  3. O. Serra
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Additional information

Supported by the Spanish Research Council (CICYT) under project TIC 90-0712 and Acción Integrada Hispanofrancesa, TIC 79B.

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Hamidoune, Y.O., Llado, A.S. & Serra, O. Minimum order of loop networks of given degree and girth. Graphs and Combinatorics 11, 131–138 (1995). https://doi.org/10.1007/BF01929482

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  • DOI: https://doi.org/10.1007/BF01929482

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