Conflict-Free Coloring of String Graphs
Chaya Keller, Alexandre Rok, Shakhar Smorodinsky

TL;DR
This paper establishes new bounds on conflict-free coloring of string graphs, linking CF-chromatic numbers to classical chromatic numbers and hypergraph properties, with tight bounds for specific classes like circle graphs.
Contribution
It provides the first general upper bounds for CF-coloring of string graphs in terms of chromatic number and hypergraph parameters, and tight bounds for circle and grounded L-shape graphs.
Findings
CF-chromatic number of string graphs is bounded by $O( ext{chromatic number}^2 \, ext{log} n)$.
For some string graph classes, CF-chromatic number is $ heta( ext{log} n)$, matching lower bounds.
Hypergraph $k$-CF-coloring can be achieved with $ ilde{O}(m^{1/(k+1)})$ colors, tight up to logs.
Abstract
Conflict-free coloring (in short, CF-coloring) of a graph is a coloring of 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 on the CF-chromatic number of any string graph with vertices in terms of the classical chromatic number . This result stands in contrast to general graphs where the CF-chromatic number can be already for bipartite graphs. (ii) For some central classes of string graphs, the CF-chromatic number is as large as…
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