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arXiv · 2511.09707

A Quasi-Polynomial Time Algorithm for 3-Coloring Circle Graphs

Ajaykrishnan E S·Robert Ganian·Daniel Lokshtanov·Vaishali Surianarayanan
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Abstract

A graph $G$ is a circle graph if it is an intersection graph of chords of a unit circle. We give an algorithm that takes as input an $n$ vertex circle graph $G$, runs in time at most $n^{O(\log n)}$ and finds a proper $3$-coloring of $G$, if one exists. As a consequence we obtain an algorithm with the same running time to determine whether a given ordered graph $(G, \prec)$ has a $3$-page book embedding. This gives a partial resolution to the well known open problem of Dujmovi\'{c} and Wood [Discret. Math. Theor. Comput. Sci. 2004], Eppstein [2014], and Bachmann, Rutter and Stumpf [J. Graph Algorithms Appl. 2024] of whether 3-Coloring on circle graphs admits a polynomial time algorithm.

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Ajaykrishnan E S, Robert Ganian, Daniel Lokshtanov, Vaishali Surianarayanan. 2025-11-12. A Quasi-Polynomial Time Algorithm for 3-Coloring Circle Graphs. https://arxiv.org/abs/2511.09707

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