## Abstract

Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let 0n,p. Let odd(G) be the number of odd degree vertices in G. Then a.a.s. the following hold:(i)G can be decomposed into ⌊Δ(G)/2⌋ cycles and a matching of size odd(G)/2.(ii)G can be decomposed into max {odd(G)/2, ⌈Δ(G)/2⌉} paths.(iii)G can be decomposed into ⌈Δ(G)/2⌉ linear forests. Each of these bounds is best possible. We actually derive (i)-(iii) from 'quasirandom' versions of our results. In that context, we also determine the edge chromatic number of a given dense quasirandom graph of even order. For all these results, our main tool is a result on Hamilton decompositions of robust expanders by Kühn and Osthus.

Original language | English |
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Pages (from-to) | 88-108 |

Number of pages | 21 |

Journal | Journal of Combinatorial Theory. Series B |

Volume | 118 |

Early online date | 28 Jan 2016 |

DOIs | |

Publication status | Published - 1 May 2016 |

## Keywords

- Cycle decomposition
- Linear arboricity
- Overfull subgraph conjecture
- Path decomposition
- Quasirandom graph
- Robust expander

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
- Computational Theory and Mathematics