For an integer t, we let Pt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_t$$\end{document} denote the t-vertex path. We write H+G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H+G$$\end{document} for the disjoint union of two graphs H and G, and for an integer r and a graph H, we write rH for the disjoint union of r copies of H. We say that a graph G is H-free if no induced subgraph of G is isomorphic to the graph H. In this paper, we study the complexity of k-coloring, for a fixed integer k, when restricted to the class of H-free graphs with a fixed graph H. We provide a polynomial-time algorithm to test if, for fixed r, a (P6+rP3)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_6+rP_3)$$\end{document}-free is three-colorable, and find a coloring if one exists. We also solve the list version of this problem, where each vertex is assigned a list of possible colors, which is a subset of {1,2,3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,2,3\}$$\end{document}. This generalizes results of Broersma, Golovach, Paulusma, and Song, and results of Klimošová, Malik, Masařík, Novotná, Paulusma, and Slívová. Our proof uses a result of Ding, Seymour, and Winkler relating matchings and hitting sets in hypergraphs. We also prove that the problem of deciding if a (P5+P2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(P_5+P_2)$$\end{document}-free graph has a k-coloring is NP-hard for every fixed k≥5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 5$$\end{document}.

中文翻译：

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列出 3-没有诱导的着色图 $$P_6+rP_3$$

对于整数 t，我们让 Pt\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek } \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_t$$\end{document} 表示 t 顶点路径。我们写 H+G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength {\oddsidemargin}{-69pt} \begin{document}$$H+G$$\end{document} 用于两个图 H 和 G 的不相交并集，对于整数 r 和图 H，我们将 rH 写为H 的 r 个副本的不相交联合。如果 G 的诱导子图没有与图 H 同构，我们说图 G 是无 H 的。在本文中，我们研究了 k 着色的复杂性，对于固定整数 k，当限制为具有固定图 H 的无 H 图类时。我们提供多项式时间算法来测试对于固定 r，a (P6+rP3)\documentclass[12pt]{minimal } \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document }$$(P_6+rP_3)$$\end{document}-free 是三种可着色的，如果存在，请找出一种着色。我们还解决了这个问题的列表版本，其中每个顶点都被分配了一个可能颜色的列表，它是 {1,2,3}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym } \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{1,2, 3\}$$\end{文档}。这概括了 Broersma、Golovach、Paulusma 和 Song 的结果，以及 Klimošová、Malik、Masařík、Novotná、Paulusma 和 Slívová 的结果。我们的证明使用了 Ding、Seymour 和 Winkler 在超图中关联匹配和命中集的结果。