Random 4-regular graphs have claw-decompositions asymptotically almost surely

In 2006, Barat and Thomassen conjectured that the edges of every planar 4-regular 4-edge connected graph can be decomposed into claws. Shortly afterward, Lai constructed a counterexample to this conjecture. Using the small subgraph conditioning method of Robinson and Wormald, we find that a random 4-regular graph has a claw-decomposition asymptotically almost surely, provided that the number of vertices is divisible by 3.