Clique Decompositions in Random Graphs via Refined Absorption

We prove that if p≥ n^-1/3+β for some β > 0, then asymptotically almost surely the binomial random graph G(n,p) has a K_3-packing containing all but at most n + O(1) edges. Similarly, we prove that if d ≥ n^2/3+β for some β > 0 and d is even, then asymptotically almost surely the random d-regular graph G_n,d has a triangle decomposition provided 3 | d · n. We also show that G(n,p) admits a fractional K_3-decomposition for such a value of p. We prove analogous versions for a K_q-packing of G(n,p) with p≥ n^-1/q+0.5+β and leave of (q-2)n+O(1) edges, for K_q-decompositions of G_n,d with (q-1) | d and d≥ n^1-1/q+0.5+β provided q| d· n, and for fractional K_q-decompositions.