  Erd\H{o}s, Gy\'arf\'as, and Pyber (1991) conjectured that every $r$-colored
complete graph can be partitioned into at most $r-1$ monochromatic components;
this is a strengthening of a conjecture of Lov\'asz (1975) in which the
components are only required to form a cover. An important partial result of
Haxell and Kohayakawa (1995) shows that a partition into $r$ monochromatic
components is possible for sufficiently large $r$-colored complete graphs.
  We start by extending Haxell and Kohayakawa's result to graphs with large
minimum degree, then we provide some partial analogs of their result for random
graphs. In particular, we show that if $p\ge \left(\frac{27\log
n}{n}\right)^{1/3}$, then a.a.s. in every $2$-coloring of $G(n,p)$ there exists
a partition into two monochromatic components, and for $r\geq 2$ if $p\ll
\left(\frac{r\log n}{n}\right)^{1/r}$, then a.a.s. there exists an $r$-coloring
of $G(n,p)$ such that there does not exist a cover with a bounded number of
components. Finally, we consider a random graph version of a classic result of
Gy\'arf\'as (1977) about large monochromatic components in $r$-colored complete
graphs. We show that if $p=\frac{\omega(1)}{n}$, then a.a.s. in every
$r$-coloring of $G(n,p)$ there exists a monochromatic component of order at
least $(1-o(1))\frac{n}{r-1}$.
