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We use a theorem by Ding, Lubetzky and Peres describing the structure of thegiant component of random graphs in the strictly supercritical regime, in orderto 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 resultto 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 randomtournaments. A $p$-random tournament on $n$ vertices is obtained from thetransitive tournament by reversing each edge independently with probability$p$. We show that for $p=\Theta(\frac 1n)$ the chromatic number of a $p$-randomtournament behaves similarly to that of a random graph with the same edgeprobability. To treat the case $p=\frac {1+\varepsilon}n$ we use theaforementioned result on MAXCUT.

On MAXCUT in strictly supercritical random graphs, and coloring of random graphs and random tournaments