Let $W_G(q_1,q_2,\ldots)$ be a weighted symmetric chromatic polynomial of a graph $G$. S. Chmutov, M. Kazarian and S. Lando in the paper arXiv:1803.09800v2 proved that the generating function $\mathcal{W}(G)$ for the polynomials $W_G(q_1,q_2,\ldots)$ is a $\tau$-function of the Kadomtsev--Petviashvili integrable hierarchy. We proved that the function $\mathcal{W}(G)$ itself is a solution of a linear integrable hierarchy. Also we described the initial conditions for the general formal $\tau$-function of the KP-hierarchy which guarantee that the $\tau$-function is a solution of a linear integrable hierarchy.