Skip to main content
×
×
Home

Decomposing Random Graphs into Few Cycles and Edges

  • DÁNIEL KORÁNDI (a1), MICHAEL KRIVELEVICH (a2) and BENNY SUDAKOV (a1)
Abstract

Over 50 years ago, Erdős and Gallai conjectured that the edges of every graph on n vertices can be decomposed into O(n) cycles and edges. Among other results, Conlon, Fox and Sudakov recently proved that this holds for the random graph G(n, p) with probability approaching 1 as n → ∞. In this paper we show that for most edge probabilities G(n, p) can be decomposed into a union of n/4 + np/2 + o(n) cycles and edges w.h.p. This result is asymptotically tight.

Copyright
References
Hide All
[1] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley.
[2] Bollobás, B. (2001) Random Graphs, second edition, Vol. 73 of Cambridge Studies in Advanced Mathematics, Cambridge University Press.
[3] Brandt, S., Broersma, H., Diestel, R. and Kriesell, M. (2006) Global connectivity and expansion: Long cycles and factors in f-connected graphs. Combinatorica 26 1736.
[4] Broder, A. Z., Frieze, A. M., Suen, S. and Upfal, E. (1996) An efficient algorithm for the vertex-disjoint paths problem in random graphs. In Proc. Seventh Annual ACM–SIAM Symposium on Discrete Algorithms: SODA '96, SIAM, pp. 261–268.
[5] Chung, F. and Lu, L. (2001) The diameter of sparse random graphs. Adv. Appl. Math. 26 257279.
[6] Conlon, D., Fox, J. and Sudakov, B. (2014) Cycle packing. Random Struct. Alg. 45 608626.
[7] Erdős, P. (1983) On some of my conjectures in number theory and combinatorics. In Proc. Fourteenth Southeastern Conference on Combinatorics, Graph Theory and Computing (Boca Raton 1983), Congr. Numer. 39 319.
[8] Erdős, P., Goodman, A. W. and Pósa, L. (1966) The representation of a graph by set intersections. Canad. J. Math. 18 106112.
[9] Knox, F., Kühn, D. and Osthus, D. Edge-disjoint Hamilton cycles in random graphs. Random Struct. Alg., to appear.
[10] Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359364.
[11] Pyber, L. (1985) An Erdős–Gallai conjecture. Combinatorica 5 6779.
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? *
×

MSC classification

Metrics

Full text views

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

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed