Decomposition of bounded degree graphs into $C_4$-free subgraphs
  We prove that every graph with maximum degree $\Delta$ admits a partition of its edges into $O(\sqrt{\Delta})$ parts (as $\Delta\to\infty$) none of which contains $C_4$ as a subgraph. This bound is sharp up to a constant factor. Our proof uses an iterated random colouring procedure. 
Ross J. Kang and Guillem Perarnau
math.CO