Linear colouring of binomial random graphs

We investigate the linear chromatic number ${\chi }_{\text{lin}}\left(G\right(n,p\left)\right)$ of the binomial random graph $G(n,p)$ on n vertices in which each edge appears independently with probability $p=p\left(n\right)$. For a graph G, ${\chi }_{\text{lin}}\left(G\right)$ is defined as the smallest k such that G admits a k-colouring with the property that every path P in G receives a colour which appears on only one vertex of P. For dense random graphs ($np\to \infty$ as $n\to \infty$), we show that asymptotically almost surely ${\chi }_{\text{lin}}\left(G\right(n,p\left)\right)\ge n(1-O({\left(np\right)}^{-1/2}\left)\right)=n(1-o(1\left)\right)$. 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.