Fractional clique decompositions of dense graphs

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Colleges, School and Institutes


For each r ≥ 4, we show that any graph G with minimum degree at least (1 − 1/(100r))|G| has a fractional Kr-decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional Kr-decomposition given by Dukes (for small r) and Barber, Kühn, Lo, Montgomery and Osthus (for large r), giving the first bound that is tight up to the constant multiple of r (seen, for example, by considering Turán graphs). In combination with work by Glock, Kühn, Lo, Montgomery and Osthus, this shows that, for any graph F with chromatic number χ(F) ≥ 4, and any ε > 0, any sufficiently large graph G with minimum degree at least (1 − 1/(100χ(F)) + ε)|G| has, subject to some further simple necessary divisibility conditions, an (exact) F-decomposition.


Original languageEnglish
Pages (from-to)779-796
Number of pages18
JournalRandom Structures and Algorithms
Issue number4
Early online date18 Oct 2018
Publication statusPublished - Jul 2019


  • Graph decomposition, Fractional graph theory, clique decompositions