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Proper Colorability of Segment Intersection Graphs

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Computing and Combinatorics (COCOON 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13595))

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Abstract

We consider the vertex proper coloring problem for highly restricted instances of geometric intersection graphs of line segments embedded in the plane. We show that, provided a graph in the class PURE-k-DIR corresponding to intersection graphs of segments lying in at most k directions with all parallel segments disjoint, and provided a k-coloring for this graph, it is NP-complete to decide if the graph admits a \(\left( k-1\right) \)-coloring \(\forall k \ge 4\). Furthermore, we show that this result holds under the constraint that all segments are of unit length in the case where \(k=4\), and under the constraint that segments have at most two distinct lengths \(\forall k \ge 5\). More generally, we establish that the problem of properly 3-coloring an arbitrary graph with m edges can be reduced in \(\mathcal {O}\left( m\right) \) time to the problem of properly 3-coloring a PURE-4-DIR graph where all segments are of unit length, yielding a method for explicit construction of hard 3-colorability instances for this graph class.

This work was supported by JSPS Kakenhi grants {20K21827, 20H05967, 21H04871}, and JST CREST Grant JPMJCR1402JST.

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Authors and Affiliations

  1. Division of Medical Data Informatics, Human Genome Center, Institute of Medical Science, University of Tokyo, 4-6-1 Shirokanedai, Minato-ku, Tokyo, 108-8639, Japan

    Robert D. Barish & Tetsuo Shibuya

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  1. Robert D. Barish
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  2. Tetsuo Shibuya
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Correspondence to Robert D. Barish .

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Editors and Affiliations

  1. Shenzhen Institute of Advanced Technology, Shenzhen, China

    Yong Zhang

  2. Harbin Institute of Technology, Harbin, China

    Dongjing Miao

  3. Technical University of Berlin, Berlin, Germany

    Rolf Möhring

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Barish, R.D., Shibuya, T. (2022). Proper Colorability of Segment Intersection Graphs. In: Zhang, Y., Miao, D., Möhring, R. (eds) Computing and Combinatorics. COCOON 2022. Lecture Notes in Computer Science, vol 13595. Springer, Cham. https://doi.org/10.1007/978-3-031-22105-7_51

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  • DOI: https://doi.org/10.1007/978-3-031-22105-7_51

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  • Online ISBN: 978-3-031-22105-7

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