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.

翻译：我们研究二项随机图 $G(n,p)$ 的线性色数 $\chi_{\text{lin}}(G(n,p))$，该图有 $n$ 个顶点，每条边以概率 $p=p(n)$ 独立出现。对于稠密随机图（当 $n \to \infty$ 时 $np \to \infty$），我们证明渐近几乎必然地有 $\chi_{\text{lin}}(G(n,p)) \ge n (1 - O( (np)^{-1/2} ) ) = n(1-o(1))$。理解次临界随机图（$np < 1$）和临界随机图（$np=1$）的线性色数量级相对容易。然而，超临界稀疏随机图（$np = c$，其中常数 $c > 1$）仍有待研究。