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

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 $G\sim G\left(n,\frac {1+\varepsilon}n\right)$intermsof$\varepsilon$. We then apply this result to prove the following conjecture by Frieze and Pegden. For every $\epsilon >0$ there exists ${\mathrm{\ell }}_{\epsilon }$ such that \whp $G\sim G(n,\frac {1+\varepsilon}n)$isnothomomorphictothecycleon$2\ell_\varepsilon+1$ 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 $p=\mathrm{\Theta }(\frac{1}{n})$ 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 $p=\frac{1+\epsilon }{n}$ we use the aforementioned result on MAXCUT.