On 3-Coloring Circle Graphs

Abstract

Given a graph $G$ with a fixed vertex order $\prec$, one obtains a circle graph $H$ whose vertices are the edges of $G$ and where two such edges are adjacent if and only if their endpoints are pairwise distinct and alternate in $\prec$. Therefore, the problem of determining whether $G$ has a $k$-page book embedding with spine order $\prec$ is equivalent to deciding whether $H$ can be colored with $k$ colors. Finding a $k$-coloring for a circle graph is known to be NP-complete for $k \geq 4$ [9] and trivial for $k \leq 2$. For $k = 3$, Unger (1992) claims an efficient algorithm that finds a 3-coloring in $O(n \log n)$ time, if it exists. Given a circle graph $H$, Unger's algorithm (1) constructs a 3-\textsc{Sat} formula $\Phi$ that is satisfiable if and only if $H$ admits a 3-coloring and (2) solves $\Phi$ by a backtracking strategy that relies on the structure imposed by the circle graph. However, the extended abstract misses several details and Unger refers to his PhD thesis (in German) for details.   In this paper we argue that Unger's algorithm for 3-coloring circle graphs is not correct and that 3-coloring circle graphs should be considered as an open problem. We show that step (1) of Unger's algorithm is incorrect by exhibiting a circle graph and its representation whose formula $\Phi$ is satisfiable but that is not 3-colorable. We further show that Unger's backtracking strategy for solving $\Phi$ in step (2) may produce incorrect results and give empirical evidence that it exhibits a runtime behaviour that is not consistent with the claimed running time.