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

TL;DR

This paper presents a quasi-polynomial time algorithm for 3-coloring circle graphs, advancing the understanding of their colorability and related embedding problems, and partially resolving a long-standing open problem.

Contribution

It introduces a quasi-polynomial time algorithm for 3-coloring circle graphs, a significant step towards polynomial algorithms for this problem.

Findings

Algorithm runs in n^{O(log n)} time

Determines 3-colorability of circle graphs efficiently

Applies to 3-page book embedding decision problem

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.