Linear Colouring of Binomial Random Graphs

We investigate the linear chromatic number $\chi_{\text{lin}}(G(n,p))$ of the binomial random graph $G(n,p)$ on $n$ vertices in which each edge appears independently with probability $p=p(n)$. For dense random graphs ($np \to \infty$ as $n \to \infty$), we show that asymptotically almost surely $\chi_{\text{lin}}(G(n,p)) \ge n (1 - O( (np)^{-1/2} ) ) = n(1-o(1))$. Understanding the order of the linear chromatic number for subcritical random graphs ($np < 1$) and critical ones ($np=1$) is relatively easy. However, supercritical sparse random graphs ($np = c$ for some constant $c > 1$) remain to be investigated.