On the exact decomposition threshold for even cycles
  A graph $G$ has a $C_k$-decomposition if its edge set can be partitioned into cycles of length $k$. We show that if $\delta(G)\geq 2|G|/3-1$, then $G$ has a $C_4$-decomposition, and if $\delta(G)\geq |G|/2$, then $G$ has a $C_{2k}$-decomposition, where $k\in \mathbb{N}$ and $k\geq 4$ (we assume $G$ is large and satisfies necessary divisibility conditions). These minimum degree bounds are best possible and provide exact versions of asymptotic results obtained by Barber, K\"uhn, Lo and Osthus. In the process, we obtain asymptotic versions of these results when $G$ is bipartite or satisfies certain expansion properties. 
Amelia Taylor
math.CO