Is it true that $\chi_L(G)=o(n)$ for almost all graphs on $n$ vertices?
Is it true that $\chi_L(G)=o(n)$ for almost all graphs on $n$ vertices?
The answer is yes: Alon [Al92] proved that in fact the random graph on $n$ vertices with edge probability $1/2$ has \[\chi_L(G) \ll \frac{\log\log n}{\log n}n\] almost surely. Alon, Krivelevich, and Sudakov [AKS99] improved this to \[\chi_L(G) \asymp \frac{n}{\log n}\] almost surely.