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 $k \geq 4$ 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 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.

翻译：给定一个具有固定顶点顺序 $\prec$ 的图 $G$，可以得到一个圆图 $H$，其顶点为 $G$ 的边，且两条边相邻当且仅当它们的端点两两不同并在 $\prec$ 中交替出现。因此，确定 $G$ 是否具有脊柱顺序 $\prec$ 的 $k$ 页书嵌入问题等价于判定 $H$ 是否可用 $k$ 种颜色染色。已知圆图的 $k$ 染色问题在 $k \geq 4$ 时是 NP-完全的，在 $k \leq 2$ 时是平凡的。对于 $k = 3$，Unger (1992) 声称存在一种高效算法，可在 $O(n \log n)$ 时间内找到三染色方案（若存在）。给定圆图 $H$，Unger 算法：(1) 构造一个 3-\textsc{Sat} 公式 $\Phi$，其可满足性当且仅当 $H$ 存在三染色；(2) 通过依赖于圆图结构约束的回溯策略求解 $\Phi$。然而，该扩展摘要遗漏了诸多细节，Unger 将具体的算法细节指向他的德文博士论文。本文中，我们论证 Unger 提出的圆图三染色算法不正确，并认为圆图的三染色问题应作为开放问题重新审视。我们通过展示一个实例圆图，其公式 $\Phi$ 可满足但该图不可三染色，证明 Unger 算法的步骤 (1) 存在错误。我们进一步指出，Unger 在步骤 (2) 中用于求解 $\Phi$ 的回溯策略可能产生错误结果，并通过实证证据表明其运行时间行为与所声称的时间复杂度不一致。