Abstract
Behzad, Chartrand and Wall conjectured that the girth of a diregular graph of ordern and outdegreer is not greater than [n /r]. This conjecture has been proved forr=2 by Behzad and forr=3 by Bermond. We prove that a digraph of ordern and halfdegree ≧4 has girth not exceeding [n / 4]. We also obtain short proofs of the above results. Our method is an application of the theory of connectivity of digraphs.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M. Behzad, G. Chartrand andC. Wall, On minimal regular digraphs with given girth,Fund. Math. 69 (1970), 227–231.
M. Behzad, Miminally 2-regular digraphs with given girth,J. Math. Soc. of Japan 25 (1973), 1–6.
C. Berge,Graphes et hypergraphes, Dunod, Paris (1970).
J. C. Bermond, 1-graphes réguliers de girth donné,Cahiers du C.E.R.O. Bruxelles 17 (1975), 123–135.
L. Caccetta andR. Häggkvist, On minimal digraphs with given girth,Congressus numerantium XXI, Utilitas Mathematica, Boca Raton (1978), 181–187.
G. A. Dirac, Extensions of Menger’s theorem,J. Lond. Math. Soc. 38 (1963), 148–161.
Y. O. Hamidoune, Sur les atomes d’un graphe orienté, C. R. Acad. Sc. Paris A284 (1977), 1253–1256.
Y. O. Hamidoune, An application of connectivity theory in graphs to factorizations of elements in finite groups,Europ. Journal of Comb. 2 (1981), 349–355.
W. Mader, Eine Eigenschaft der Atome endlicher Graphen,Arch. Math. 22 (1971), 333–336.
K. Menger, Zur allgemeinen Kurventheorie,Fundamenta Math. 1 (1927), 96–115.