Abstract
In this paper we consider the random r-uniform r-partite hypergraph model H(n 1, n 2, ···, n r; n, p) which consists of all the r-uniform r-partite hypergraphs with vertex partition {V 1, V 2, ···, V r} where |V i| = n i = n i(n) (1 ≤ i ≤ r) are positive integer-valued functions on n with n 1 +n 2 +···+n r = n, and each r-subset containing exactly one element in V i (1 ≤ i ≤ r) is chosen to be a hyperedge of H p ∈ H (n 1, n 2, ···, n r; n, p) with probability p = p(n), all choices being independent. Let
and
be the maximum and minimum degree of vertices in V 1 of H, respectively;
,
be the number of vertices in V 1 of H with degree d, at least d, at most d, and between c and d, respectively. In this paper we obtain that in the space H(n 1, n 2, ···, n r; n, p),
all have asymptotically Poisson distributions. We also answer the following two questions. What is the range of p that there exists a function D(n) such that in the space H(n 1, n 2, ···, n r; n, p),
? What is the range of p such that a.e., H p ∈ H (n 1, n 2, ···, n r; n, p) has a unique vertex in V 1 with degree
? Both answers are p = o (log n 1/N), where
. The corresponding problems on
also are considered, and we obtained the answers are p ≤ (1 + o(1))(log n 1/N) and p = o (log n 1/N), respectively.
Similar content being viewed by others
References
Bollobás, B. The distribution of the maximum degree of a random graph. Discrete Mathematics, 32: 201–203 (1980)
Bollobás, B. Degree sequences of random graphs. Discrete Mathematics, 33: 1–19 (1981)
Bollobás, B. Vertices of given degree in a random graph. Journal of Graph Theory, 6: 147–155 (1982)
Bollobás, B. Random graphs (2nd). Cambridge University Press, Cambridge, 2001
Erdös, P. Graph theory and probability. Canad. J. Math., 11: 34–38 (1959)
Erdös, P., Wilson, R.J. On the chromatic index of almost all graphs. Journal of Combinatorial Theory B, 23: 255–257 (1977)
Feller, W. An introduction to probability theory and its applications, Vols. I and II.John Wiley and Sons, New York, London, Sydney, 1966
Ivchenko, G.I. On the asymptotic behavior of the degrees of vertices in a random graph. Theory of Probability and its Applications, 18: 188–195 (1973)
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported in part by the National Natural Science Foundation of China under Grant No. 11401102, 11271307 and 11101086, Fuzhou university of Science and Technology Development Fund No. 2014-XQ-29.
Rights and permissions
About this article
Cite this article
Chen, Al., Li, H. & Zhang, Fj. The maximum and minimum degree of the random r-uniform r-partite hypergraphs. Acta Math. Appl. Sin. Engl. Ser. 32, 867–874 (2016). https://doi.org/10.1007/s10255-016-0606-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-016-0606-5