  We use a theorem by Ding, Lubetzky and Peres describing the structure of the
giant component of random graphs in the strictly supercritical regime, in order
to determine the typical size of MAXCUT of $G\sim G\left(n,\frac
{1+\varepsilon}n\right)$ in terms of $\varepsilon$. We then apply this result
to prove the following conjecture by Frieze and Pegden. For every
$\varepsilon>0$ there exists $\ell_\varepsilon$ such that \whp $G\sim G(n,\frac
{1+\varepsilon}n)$ is not homomorphic to the cycle on $2\ell_\varepsilon+1$
vertices. We also consider the coloring properties of biased random
tournaments. A $p$-random tournament on $n$ vertices is obtained from the
transitive tournament by reversing each edge independently with probability
$p$. We show that for $p=\Theta(\frac 1n)$ the chromatic number of a $p$-random
tournament behaves similarly to that of a random graph with the same edge
probability. To treat the case $p=\frac {1+\varepsilon}n$ we use the
aforementioned result on MAXCUT.
