Chromatic discrepancy of locally $s$-colourable graphs

The chromatic discrepancy of a graph $G$, denoted $\phi(G)$, is the least over all proper colourings $\sigma$ of $G$ of the greatest difference between the number of colours $|\sigma(V(H))|$ spanned by an induced subgraph $H$ of $G$ and its chromatic number $\chi(H)$. We prove that the chromatic discrepancy of a triangle-free graph $G$ is at least $\chi(G)-2$. This is best possible and positively answers a question raised by Aravind, Kalyanasundaram, Sandeep, and Sivadasan. More generally, we say that a graph $G$ is locally $s$-colourable if the closed neighbourhood of any vertex $v\in V(G)$ is properly $s$-colourable; in particular, a triangle-free graph is locally $2$-colourable. We conjecture that every locally $s$-colourable graph $G$ satisfies $\phi(G) \geq \chi(G)-s$, and show that this would be almost best possible. We prove the conjecture when $\chi(G) \le 11s/6$, and as a partial result towards the general case, we prove that every locally $s$-colourable graph $G$ satisfies $\phi(G) \geq \chi(G) - s\ln \chi(G)$. If the conjecture holds, it implies in particular, for every integer $\ell\geq 2$, that any graph $G$ without any copy of $C_{\ell+1}$, the cycle of length $\ell+1$, satisfies $\phi(G) \geq \chi(G) - \ell$. When $\ell \ge 3$ and $G\neq K_\ell$, we conjecture that we actually have $\phi(G)\ge \chi(G) - \ell + 1$, and prove it in the special case $\ell = 3$ or $\chi(G) \le 5\ell/3$. In general, we further obtain that every $C_{\ell+1}$-free graph $G$ satisfies $\phi(G) \geq \chi(G) - O_{\ell}(\ln \ln \chi(G))$. We do so by determining an almost tight bound on the chromatic number of balls of radius at most $\ell/2$ in $G$, which could be of independent interest.