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The '''1-factorization conjecture'''<ref>{{harvtxt|Chetwynd|Hilton|1985}}. {{harvtxt|Niessen|1994}}. {{harvtxt|Perkovic|Reed|1997}}. [[#West1FC|West]].</ref> is a long-standing [[conjecture]] that states that ''k'' ≈ ''n'' is sufficient. In precise terms, the conjecture is:
* If ''n'' is odd and ''k'' ≥ ''n'', then ''G'' is 1-factorable. If ''n'' is even and ''k'' ≥ ''n'' − 1 then ''G'' is 1-factorable.
The [[overfull conjecture]] implies the 1-factorization conjecture.
The conjecture was confirmed by Csaba, Kühn, Lo, Osthus and Treglown for sufficiently large ''n''.<ref>{{harvtxt|Csaba|Kühn|Lo|Osthus|Treglown|2016}}.</ref> <ref name="wanless">
{{Citation
| last1 = Csaba
| first1 = Béla
| last2 = Kühn
| first2 = Daniela
| last3 = Lo
| first3 = Allan
| last4 = Osthus
| first4 = Deryk
| last5 = Treglown
| first5 = Andrew
| title = Proof of the 1-factorization and Hamilton decomposition conjectures
| journal = Memoirs of the American Mathematical Society
| date = June 2016
| doi = 10.1090/memo/1154
}}
</ref>
===Perfect 1-factorization===
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