Conflict-Free Coloring of String Graphs

Conflict-free coloring (in short, CF-coloring) of a graph $G = (V,E)$ is a coloring of $V$ such that the neighborhood of each vertex contains a vertex whose color differs from the color of any other vertex in that neighborhood. Bounds on CF-chromatic numbers have been studied both for general graphs and for intersection graphs of geometric shapes. In this paper we obtain such bounds for several classes of string graphs, i.e., intersection graphs of curves in the plane: (i) We provide a general upper bound of $O(\chi(G)^2 \log n)$ on the CF-chromatic number of any string graph $G$ with $n$ vertices in terms of the classical chromatic number $\chi(G)$. This result stands in contrast to general graphs where the CF-chromatic number can be $\Omega(\sqrt{n})$ already for bipartite graphs. (ii) For some central classes of string graphs, the CF-chromatic number is as large as $\Theta(\sqrt{n})$, which is the upper bound for any graph even in the non-geometric context. For several such classes (e.g., intersection graphs of frames) we prove a tight bound of $\Theta(\log n)$ with respect to the notion of $k$-CF-coloring (in which the punctured neighborhood of each vertex contains a color that appears at most $k$ times), for a small constant $k$. (iii) We obtain a general upper bound on the $k$-CF-chromatic number of arbitrary hypergraphs: Any hypergraph with $m$ hyperedges can be $k$-CF colored with $\tilde{O}(m^{\frac{1}{k+1}})$ colors. This bound, which extends a bound of Pach and Tardos (2009), is tight for some string graphs, up to a logarithmic factor. (iv) Our fourth result concerns circle graphs in which coloring problems are motivated by VLSI designs. We prove a tight bound of $\Theta(\log n)$ on the CF-chromatic number of circle graphs, and an upper bound of $O(\log^{3} n)$ for a wider class that contains circle graphs, namely, intersection graphs of grounded L-shapes.