Abstract
In this paper it is deduced the number ofs-paths (s-cycles) havingk edges in common with a fixeds-path (s-cycle) of the complete graphK n (orK* n for directed graphs).
It is also proved that the number of the common edges of twos-path (s-cycles) randomly chosen from the set ofs-paths (s-cycles) ofK n (respectivelyK* n ), are random variables, distributed asymptotically in accordance with the Poisson law whenever\(\mathop {\lim }\limits_{n \to \infty } \) s/n exists, thus extending a result by Baróti.
Some estimations of the numbers of paths and cycles for almost all graphs and digraphs are made by applying Chebyshev’s inequality.
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References
G. Baróti, On the number of certain hamilton circuits of a complete graph,Periodica Math. Hung.,3 (1973), 135–139.
L. Lovász,Combinatorial problems and exercices, Akadémiai Kiadó, Budapest (1979).
J. W. Moon, Counting labelled trees,Canadian Math. Monographs 1 (1970), Canadian Math. Congress, W. Clowes and Sons, London and Beccles.
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Tomescu, I. On the number of paths and cycles for almost all graphs and digraphs. Combinatorica 6, 73–79 (1986). https://doi.org/10.1007/BF02579411
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DOI: https://doi.org/10.1007/BF02579411