Skip to main content

On Edge-Disjoint Spanning Trees in a Randomly Weighted Complete Graph


Assume that the edges of the complete graph Kn are given independent uniform [0, 1] weights. We consider the expected minimum total weight μk of k ⩽ 2 edge-disjoint spanning trees. When k is large we show that μk k 2. Most of the paper is concerned with the case k = 2. We show that m 2 tends to an explicitly defined constant and that μ 2 ≈ 4.1704288. . . .

Hide All
[1] Aldous D. (1992) Asymptotics in the random assignment problem. Probab. Theory Rel. Fields 93 507534.
[2] Aldous D. (2001) The ζ(2) limit in the random assignment problem. Random Struct. Alg. 4 381418.
[3] Avram F. and Bertsimas D. (1992) The minimum spanning tree constant in geometrical probability and under the independent model: A unified approach. Ann. Appl. Probab. 2 113130.
[4] Beveridge A., Frieze A. M. and McDiarmid C. J. H. (1998) Minimum length spanning trees in regular graphs. Combinatorica 18 311333.
[5] Bollobás B. (1980) A probabilistic proof of an asymptotic formula for the number of labelled graphs. Europ. J. Combin. 1 311316.
[6] Cain J. A., Sanders P. and Wormald N. (2007) The random graph threshold for k-orientability and a fast algorithm for optimal multiple-choice allocation. In SODA 2007: Proc. 18th Annual ACM–SIAM Symposium on Discrete Algorithms, SIAM, pp. 469476.
[7] Cooper C., Frieze A. M., Ince N., Janson S. and Spencer J. (2016) On the length of a random minimum spanning tree. Combin. Probab. Comput. 25, 89107.
[8] Durrett R. (1991) Probability: Theory and Examples, Wadsworth & Brooks/Cole.
[9] Fenner T. I. and Frieze A. M. (1982) On the connectivity of random m-orientable graphs and digraphs. Combinatorica 2 347359.
[10] Frieze A. M. (1985) On the value of a random minimum spanning tree problem. Discrete Appl. Math. 10 4756.
[11] Frieze A. M. (1986) On large matchings and cycles in sparse random graphs. Discrete Math. 59 243256.
[12] Frieze A. M. (2004) On random symmetric travelling salesman problems. Math. Oper. Res. 29 878890.
[13] Frieze A. M. and Grimmett G. R. (1985) The shortest path problem for graphs with random arc-lengths. Discrete Appl. Math. 10 5777.
[14] Frieze A. M. and McDiarmid C. J. H. (1989) On random minimum length spanning trees. Combinatorica 9 363374.
[15] Frieze A. M., Ruszinko M. and Thoma L. (2000) A note on random minimum length spanning trees. Electron. J. Combin. 7 R41.
[16] Gao P., Pérez-Giménez X. and Sato C. M. (2014) Arboricity and spanning-tree packing in random graphs with an application to load balancing. Extended abstract published in SODA 2014, pp. 317–326.
[17] Janson S. (1995) The minimal spanning tree in a complete graph and a functional limit theorem for trees in a random graph. Random Struct. Alg. 7 337355.
[18] Janson S. (1999) One, two and three times log n/n for paths in a complete graph with random weights. Combin. Probab. Comput. 8 347361.
[19] Janson S. and Łuczak M. J. (2007) A simple solution to the k-core problem. Random Struct. Alg. 30 5062.
[20] Karp R. M. (1979) A patching algorithm for the non-symmetric traveling salesman problem. SIAM J. Comput. 8 561573.
[21] Kordecki W. and Lyczkowska-Hanćkowiak A. (2013) Exact expectation and variance of minimal basis of random matroids. Discussiones Mathematicae Graph Theory 33 277288.
[22] Linusson S. and Wästlund J. (2004) A proof of Parisi's conjecture on the random assignment problem. Probab. Theory Rel. Fields 128 419440.
[23] Łuczak T. (1991) Size and connectivity of the k-core of a random graph. Discrete Math. 91 6168.
[24] Nair C., Prabhakar B. and Sharma M. (2005) Proofs of the Parisi and Coppersmith–Sorkin random assignment conjectures. Random Struct. Alg. 27 413444.
[25] Nash-Williams C. St. J. A. (1961) Edge-disjoint spanning trees of finite graphs. J. London Math. Soc. 36 445450.
[26] Nash-Williams C. St. J. A. (1964) Decomposition of finite graphs into forests. J. London Math. Soc. 39 12.
[27] Oxley J. (1992) Matroid Theory, Oxford University Press.
[28] Penrose M. (1998) Random minimum spanning tree and percolation on the n-cube. Random Struct. Alg. 12 6382.
[29] Pittel B., Spencer J. and Wormald N. (1996) Sudden emergence of a giant k-core in a random graph. J. Combin. Theory Ser. B 67 111151.
[30] Steele J. M. (1987) On Frieze's ζ(3) limit for lengths of minimal spanning trees. Discrete Appl. Math. 18 99103.
[31] Wästlund J. (2009) An easy proof of the ζ(2) limit in the random assignment problem. Electron. Comm. Probab. 14 261269.
[32] Wästlund J. (2010) The mean field traveling salesman and related problems. Acta Math. 204 91150.
[33] Welsh D. J. A. (1976) Matroid Theory, Academic Press.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 15 *
Loading metrics...

Abstract views

Total abstract views: 99 *
Loading metrics...

* Views captured on Cambridge Core between 9th October 2017 - 23rd January 2018. This data will be updated every 24 hours.