Chromatic number of Erdos-Renyi random graphs $G(m,n)$ for $n=4$ and $m=5$

Question

In Erdos-Renyi random graphs $G(n,m)$, set $n=4$ and $m=5$.

The question is as follows:

What is the probability for to having Chromatic number exactly 2 in the case of $G(4,5)$; in other words what is $Pr(\chi (G(4,5)) = 2)=$?

If we know that the chromatic number of a graph $G$ is the least number $k$ for which $G$ is $k-$colorable.

Hint: maybe it will be very useful to try to see how do the graphs looks like here in $G(4,5)$.

Thanks!

Answer 1

HINT: The complete graph on $4$ vertices has $\binom{4}2=6$ edges, so any graph on $4$ vertices with $5$ edges is just $K_4$ minus one edge.