Fractional cycle decompositions in hypergraphs

We prove that for any integer and , there is an integer such that any k ‐uniform hypergraph on n vertices with minimum codegree at least has a fractional decomposition into (tight) cycles of length ( ‐cycles for short) whenever and n is large in terms of . This is essentially tight. This immediately yields also approximate integral decompositions for these hypergraphs into ‐cycles. Moreover, for graphs this even guarantees integral decompositions into ‐cycles and solves a problem posed by Glock, Kühn, and Osthus. For our proof, we introduce a new method for finding a set of ‐cycles such that every edge is contained in roughly the same number of ‐cycles from this set by exploiting that certain Markov chains are rapidly mixing.