Odd Cycle Decompositions

We determine the asymptotic minimum-degree threshold for decompositions into cycles of every fixed odd length. We prove that, for each odd integer $\mathrm{\ell }\ge 3$, every $\text{eps}>0$ and sufficiently large $n$, every $n$-vertex $C_{\mathrm{\ell }}$-divisible graph $G$ with minimum degree $\delta (G)\ge (\mathrm{\ell }\mathrm{/}(2\mathrm{\ell }-2)+\text{eps})n$ has a $C_{\mathrm{\ell }}$-decomposition, where $C_{\mathrm{\ell }}$ denotes the cycle of length $\mathrm{\ell }$. The constant is best possible. This settles the remaining cases $\mathrm{\ell }=5,7,\dots ,71$ and completes the determination of the thresholds for all odd cycles. The proof combines the condensation approach of Bryant et al. with a sharp weighted-triangle fractional decomposition theorem. Its key step is a rescaling of the dual edge weights that allows us to apply a quantitative discharging estimate from the breakthrough of Delcourt and Postle on Nash–Williams’ triangle decomposition conjecture.