Chromatic discrepancy of locally $s$-colourable graphs

The chromatic discrepancy of a graph $G$, denoted $ϕ(G)$, is the least over all proper colourings $σ$ of $G$ of the greatest difference between the number of colours $|σ(V(H))|$ spanned by an induced subgraph $H$ of $G$ and its chromatic number $χ(H)$. We prove that the chromatic discrepancy of a triangle-free graph $G$ is at least $χ(G)-2$. 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 $ϕ(G) \geq χ(G)-s$, and show that this would be almost best possible. We prove the conjecture when $χ(G) \le 11s/6$, and as a partial result towards the general case, we prove that every locally $s$-colourable graph $G$ satisfies $ϕ(G) \geq χ(G) - s\ln χ(G)$. 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 $ϕ(G) \geq χ(G) - \ell$. When $\ell \ge 3$ and $G\neq K_\ell$, we conjecture that we actually have $ϕ(G)\ge χ(G) - \ell + 1$, and prove it in the special case $\ell = 3$ or $χ(G) \le 5\ell/3$. In general, we further obtain that every $C_{\ell+1}$-free graph $G$ satisfies $ϕ(G) \geq χ(G) - O_{\ell}(\ln \ln χ(G))$. 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.