Proper colorability of segment intersection graphs

We consider the vertex proper coloring problem for highly restricted instances of geometric intersection graphs of line segments embedded in the plane. Provided a graph in the class UNIT-PURE-k-DIR, corresponding to intersection graphs of unit length segments lying in at most k directions with all parallel segments disjoint, and provided explicit coordinates for segments whose intersections induce the graph, we show for \(k = 4\) that it is NP-complete to decide if a proper 3-coloring exists, and moreover, \(\#P\)-complete under many-one counting reductions to determine the number of such colorings. In addition, under the more relaxed constraint that segments have at most two distinct lengths, we show these same hardness results hold for finding and counting proper \(\left( k-1\right) \)-colorings for every \(k \ge 5\). More generally, we establish that the problem of proper 3-coloring an arbitrary graph with m edges can be reduced in \({\mathcal {O}}\left( m^2\right) \) time to the problem of proper 3-coloring a UNIT-PURE-4-DIR graph. This can then be shown to imply that no \(2^{o\left( \sqrt{n}\right) }\) time algorithm can exist for proper 3-coloring PURE-4-DIR graphs under the Exponential Time Hypothesis (ETH), and by a slightly more elaborate construction, that no \(2^{o\left( \sqrt{n}\right) }\) time algorithm can exist for counting the such colorings under the Counting Exponential Time Hypothesis (#ETH). Finally, we prove an NP-hardness result for the optimization problem of finding a maximum order proper 3-colorable induced subgraph of a UNIT-PURE-4-DIR graph.