Fractional clique decompositions of dense graphs and hypergraphs

Ben Barber, Daniela Kühn, Allan Lo, Richard Montgomery and Deryk Osthus, Journal of Combinatorial Theory, Series B PDF

Together with Daniela Kühn, Allan Lo and Deryk Osthus I proved that for every graph F there is a constant c_F < 1 such that every “F-divisible” graph G on n vertices with minimum degree at least (c_F + o(1))n has an F-decomposition. In practice, the current obstacle to improving the bounds on c_F is usually our knowledge of another quantity, the fractional decomposition threshold for cliques.

A graph G has a fractional F-decomposition if we can assign a non-negative weight to each copy of F in G such that the total weight of the copies of F containing each fixed edge of G is exactly 1. We prove that every graph with minimum degree at least (1-1/10000r^{3/2})n has a fractional K_r-decomposition. This greatly improves the previous bound of (1-2/9r^4)n for large r. We also prove a similar result for hypergraphs.

The proof begins with an approximate fractional K_r-decomposition obtained by weighting every r-clique in our graph equally. We then use small gadgets to make local adjustments to the total weight over each edge until we end up with a genuine fractional K_r-decomposition.

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