We prove that if $p\ge n^{ -\frac{ 1}{ 3}+\beta}$ for some $\beta > 0$, 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 $d \ge n^{ \frac{ 2}{ 3}+\beta}$ for some $\beta > 0$ and $d$ is even, then asymptotically almost surely the random $d$-regular graph $G_{ n,d}$ has a triangle decomposition provided $3 \mid d \cdot n$. 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 $p\ge n^{ -\frac{ 1}{ q+0.5}+\beta}$ and leave of $(q-2)n+O(1)$ edges, for $K_q$-decompositions of $G_{ n,d}$ with $(q-1)~|~d$ and $d\ge n^{ 1-\frac{ 1}{ q+0.5}+\beta}$ provided $q\mid d\cdot n$, and for fractional $K_q$-decompositions.
Categories: math.CO