On the independence number of intersection graphs of axis-parallel segments
DOI:
https://doi.org/10.20382/jocg.v14i1a5Abstract
We prove that for any triangle-free intersection graph of $n$ axis-parallel line segments in the plane, the independence number $\alpha$ of this graph is at least $\alpha \ge n/4 + \Omega(\sqrt{n})$. We complement this with a construction of a graph in this class satisfying $\alpha \le n/4 + c \sqrt{n}$ for an absolute positive constant $c$, which demonstrates the optimality of our result.
Wegner's conjecture states that the clique covering number of the intersection graph of axis-parallel rectangles can be bounded by twice its independence number. As a consequence of our results, we show that the multiplicative constant in Wegner's conjecture is tight already for the highly restricted case of axis-parallel segments, even with the assumption of triangle-freeness. Still, for this class, we improve on Wegner's bound by an additive term of order $\sqrt{\alpha}$.
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Copyright (c) 2023 Marco Caoduro, Jana Cslovjecsek, Michal Pilipczuk, Karol Wegrzycki
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