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

TL;DR

The paper addresses the open problem of polynomial-time 3-coloring circle graphs by delivering a quasi-polynomial-time algorithm with runtime $n^{O(\log n)}$ for Circle Graph List 3-Coloring, and shows that the same running time applies to determining 3-page book embeddings for ordered graphs. The approach relies on a framework of circle partitions and a branching scheme on 4-partitions to decompose the input into smaller, independently solvable subinstances, governed by a recurrence that yields the desired time bound. A key technical contribution is the construction of a polynomial-size family of fully-separated instances via a reduction rule (RR) and carefully designed properties, ensuring correctness and enabling recursive solving. The results provide strong evidence that 3-coloring circle graphs is unlikely to be NP-hard, offer a constructive method, and open avenues toward a polynomial-time algorithm or quasi-polynomial lower bounds for this classical problem and its connection to 3-page book embeddings.

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 Dujmović 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.

Figures (6)

• Figure 1: Correspondence between 3-page book embedding and circle graph 3-coloring (Source BachmannRS24)
• Figure 2: Subfigure \ref{['subfig:prelima']} shows a unit circle with arcs $ab$, $bc$, $cd$, and $da$ forming a circle partition $\mathcal{P}$, where $L(\mathcal{P}) = ab$, $T(\mathcal{P}) = bc$, $R(\mathcal{P}) = cd$, and $B(\mathcal{P}) = da$. Also, $x_1x_2$ and $y_1y_2$ are $L(\mathcal{P})$–$R(\mathcal{P})$ and $R(\mathcal{P})$–$R(\mathcal{P})$ chords, respectively. Subfigure \ref{['subfig:prelimb']} depicts a chord representation of a Circle Graph List 3-Coloring instance $I$ with a circle partition $\mathcal{P} = (L,T,R,B)$, where lists are indicated using red triangles, green squares, and blue circles.
• Figure 3: Subfigure \ref{['subfig:2a']} is a chord diagram corresponding to a fully-separated instance $(I,\mathcal{P})$ (arcs $T(\mathcal{P})$ and $B(\mathcal{P})$ are not shown since they are empty). Subfigures \ref{['subfig:2b']} and \ref{['subfig:2c']} show chord diagrams corresponding to the subinstances $I_{L(\mathcal{P})}$ and $I_{R(\mathcal{P})}$ of $I$ respectively.
• Figure 4: In Subfigure \ref{['subfig:0a']}, as all $L$-$R$ chords intersect all $T$-$B$ chords, Observation \ref{['obs:intersecting-chords']} implies that, in any valid 3-coloring, at least one of these two sets contain chords of at most one color. In line with this, all $L$-$R$ chords in Subfigure \ref{['subfig:0a']}, are colored red. Subfigure \ref{['subfig:0b']} shows the instance after applying RR to all $L$-$R$ chords in the instance, while Subfigure \ref{['subfig:0c']} has the instance after RR is applied exhaustively. We use dotted lines to depict deleted chords.
• Figure 5: Subfigure \ref{['subfig:3a']} shows a Circle Graph List 3-Coloring instance $I$ and a circle partition $\mathcal{P}=(L,T,R,B)$ with equal number of chord endpoints in each arc. Subfigure \ref{['subfig:3c']} shows a semiseparated instance $(I',\mathcal{P})$, in the family obtained after applying Lemma \ref{['lem:semi-separated']} to $(I,\mathcal{P})$. Lemma \ref{['lem:semi-separated']} internally calls Lemma \ref{['lem:helper-eliminate-chords']} on $I$, $L$-$R$ chords, $T$-$B$ chords. This instance is obtained when Lemma \ref{['lem:helper-eliminate-chords']} eliminates $L$-$R$ chords by coloring them red.

Theorems & Definitions (15)

• Theorem 3.1
• Lemma 3.3
• proof : Proof of Theorem \ref{['thm:main_thm']}
• Claim 3.3.1
• proof : Proof of the Claim.
• proof
• Lemma 3.6
• proof
• Lemma 3.7
• proof