Fractional clique decompositions of dense graphs

• Published in: Random structures & algorithms Vol. 54; no. 4; pp. 779 - 796
• Main Author: Montgomery, Richard
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc 01.07.2019
• Subjects: clique decompositions; Decomposition; Economic aid; Fractional graph theory; Graph decompositions; Graphs
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/rsa.20809

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.