论文标题
List-3彩色有序图，带有禁止的诱导子图
List-3-Coloring ordered graphs with a forbidden induced subgraph
论文作者
论文摘要
The List-3-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,2,3\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $ϕ$ with $ϕ(v)\in L(v)$ for every vertex $v$ of $G$, and the $3$-Coloring Problem is the list- $ 3 $ - 颜色的问题在使用$ l（v）= \ {1,2,3 \} $的实例上，每个顶点$ v $ of $ g $。列表-3 $颜色的问题是一个经典的NP完整问题，众所周知，虽然仅限于$ h $ free Graphs（意味着没有诱发的固定图形$ H $的图形同构图），但除非$ h $是np-potherte，否则$ h $是对路径的引起的子段的同量。但是，目前的艺术状态远非证明这足以满足多项式时间算法。实际上，$ p_8 $ free Graphs（$ p_8 $表示八个vertex路径）上$ 3 $颜色问题的复杂性是未知的。在这里，我们考虑列表的变体-3美元的颜色问题，称为有序图表-3 $ - 颜色的问题，其中输入是有序的图，也就是说，图形以及其顶点集合上的线性顺序。对于有序图$ g $和$ h $，我们说$ g $是$ h $ - 如果$ h $不是同构的$ g $同构的同构，则保留线性订单。我们证明，假设$ h $是有序的图表，这是订购的图表清单几乎完整的二分法-3 $颜色的问题仅限于$ h $ free订购的图形。特别是，我们表明，如果$ h $最多具有一个边缘，则可以在多项式时间内解决问题，并且如果$ h $至少具有三个边缘，则可以保持NP填充。此外，如果$ h $完全具有两个边缘，那么当两个边缘的两个边缘共享时，我们就会给出一个完整的二分法，当两个$ h $的两个边缘没有共享结局时，就会证明几个NP完整性结果，将开放式箱子缩小到三个非常特殊类型的两种距离订购的图形。
The List-3-Coloring Problem is to decide, given a graph $G$ and a list $L(v)\subseteq \{1,2,3\}$ of colors assigned to each vertex $v$ of $G$, whether $G$ admits a proper coloring $ϕ$ with $ϕ(v)\in L(v)$ for every vertex $v$ of $G$, and the $3$-Coloring Problem is the List-$3$-Coloring Problem on instances with $L(v)=\{1,2,3\}$ for every vertex $v$ of $G$. The List-$3$-Coloring Problem is a classical NP-complete problem, and it is well-known that while restricted to $H$-free graphs (meaning graphs with no induced subgraph isomorphic to a fixed graph $H$), it remains NP-complete unless $H$ is isomorphic to an induced subgraph of a path. However, the current state of art is far from proving this to be sufficient for a polynomial time algorithm; in fact, the complexity of the $3$-Coloring Problem on $P_8$-free graphs (where $P_8$ denotes the eight-vertex path) is unknown. Here we consider a variant of the List-$3$-Coloring Problem called the Ordered Graph List-$3$-Coloring Problem, where the input is an ordered graph, that is, a graph along with a linear order on its vertex set. For ordered graphs $G$ and $H$, we say $G$ is $H$-free if $H$ is not isomorphic to an induced subgraph of $G$ with the isomorphism preserving the linear order. We prove, assuming $H$ to be an ordered graph, a nearly complete dichotomy for the Ordered Graph List-$3$-Coloring Problem restricted to $H$-free ordered graphs. In particular, we show that the problem can be solved in polynomial time if $H$ has at most one edge, and remains NP-complete if $H$ has at least three edges. Moreover, in the case where $H$ has exactly two edges, we give a complete dichotomy when the two edges of $H$ share an end, and prove several NP-completeness results when the two edges of $H$ do not share an end, narrowing the open cases down to three very special types of two-edge ordered graphs.
