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We address recently proposed chromatic versions of the classic Art GalleryProblem. Assume a simple polygon $P$ is guarded by a finite set of point guardsand each guard is assigned one of $t$ colors. Such a chromatic guarding is saidto be conflict-free if each point $p\in P$ sees at least one guard with aunique color among all guards visible from $p$. The goal is to establish boundson the function $\chi_{cf}(n)$ of the number of colors sufficient to guaranteethe existence of a conflict-free chromatic guarding for any $n$-vertex polygon.B\"artschi and Suri showed $\chi_{cf}(n)\in O(\log n)$ (Algorithmica, 2014) forsimple orthogonal polygons and the same bound applies to general simplepolygons (B\"artschi et al., SoCG 2014). In this paper, we assume ther-visibility model instead of standard line visibility. Points $p$ and $q$ inan orthogonal polygon are r-visible to each other if the rectangle spanned bythe points is contained in $P$. For this model we show $\chi_{cf}(n)\inO(\log\log n)$ and $\chi_{cf}(n)\in \Omega(\log\log n /\log\log\log n)$. Mostinterestingly, we can show that the lower bound proof extends to guards withline visibility. To this end we introduce and utilize a novel discretecombinatorial structure called multicolor tableau. This is the firstnon-trivial lower bound for this problem setting.Furthermore, for the strongchromatic version of the problem, where all guards r-visible from a point musthave distinct colors, we prove a $\Theta(\log n)$-bound. Our results can beinterpreted as coloring results for special geometric hypergraphs.

Almost Tight Bounds for Conflict-Free Chromatic Guarding of Orthogonal Galleries