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The chromatic discrepancy of graphs

For a proper vertex coloring $c$ of a graph $G$, let $φ_c(G)$ denote the maximum, over all induced subgraphs $H$ of $G$, the difference between the chromatic number $χ(H)$ and the number of colors used by $c$ to color $H$. We define the chromatic discrepancy of a graph $G$, denoted by $φ(G)$, to be the minimum $φ_c(G)$, over all proper colorings $c$ of $G$. If $H$ is restricted to only connected induced subgraphs, we denote the corresponding parameter by $\hatφ(G)$. These parameters are aimed at studying graph colorings that use as few colors as possible in a graph and all its induced subgraphs. We study the parameters $φ(G)$ and $\hatφ(G)$ and obtain bounds on them. We obtain general bounds, as well as bounds for certain special classes of graphs including random graphs. We provide structural characterizations of graphs with $φ(G) = 0$ and graphs with $\hatφ(G) = 0$. We also show that computing these parameters is NP-hard.