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

2025-11-12

AI Summary

This paper addresses the long-standing open problem of 3-coloring circle graphs. We present the first quasipolynomial-time algorithm that, for any $n$-vertex circle graph, decides 3-colorability in $n^{O(log n)}$ time and constructs a valid 3-coloring if one exists. Our approach leverages an ordered recursive decomposition of the chord intersection structure inherent to circle graphs, combining dynamic programming with divide-and-conquer techniques to circumvent traditional exponential-time bottlenecks. This result improves the best-known upper bound for 3-coloring circle graphs from $2^{O(n)}$ to quasipolynomial time, making significant progress toward the open question of whether a polynomial-time algorithm exists. Moreover, our algorithm directly solves the 3-page book embedding existence problem for circle graphs—yielding the first quasipolynomial-time solution for this fundamental graph layout problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Applications of cryptography

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.

Problem

• Develops quasi-polynomial algorithm for 3-coloring circle graphs
• Solves 3-page book embedding problem for ordered graphs
• Partially resolves open problem about polynomial-time 3-coloring

Innovation

• Quasi-polynomial algorithm for circle graph coloring
• Transforms chord intersection graphs into colorings
• Solves 3-page book embedding via coloring reduction