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Decomposing Random Graphs into Few Cycles and Edges

Part of: Graph theory

Published online by Cambridge University Press:  29 December 2014

DÁNIEL KORÁNDI ,
MICHAEL KRIVELEVICH  and
BENNY SUDAKOV
DÁNIEL KORÁNDI
Affiliation:
Department of Mathematics, ETH, Rämistrasse 101, 8092 Zurich, Switzerland (e-mail: daniel.korandi@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
MICHAEL KRIVELEVICH
Affiliation:
School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801, Israel (e-mail: krivelev@post.tau.ac.il)
BENNY SUDAKOV
Affiliation:
Department of Mathematics, ETH, Rämistrasse 101, 8092 Zurich, Switzerland (e-mail: daniel.korandi@math.ethz.ch, benjamin.sudakov@math.ethz.ch)
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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.

MSC classification

Primary: 05C38: Paths and cycles
Secondary: 05C80: Random graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 6 , November 2015 , pp. 857 - 872
Copyright
Copyright © Cambridge University Press 2014 

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