A proper t-coloring of a graph G is a mapping \documentclass[12pt]{minimal}
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\begin{document}$${\varphi: V(G) \rightarrow [1, t]}$$\end{document} such that \documentclass[12pt]{minimal}
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\begin{document}$${\varphi(u) \neq \varphi(v)}$$\end{document} if u and v are adjacent vertices, where t is a positive integer. The chromatic number of a graph G, denoted by \documentclass[12pt]{minimal}
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\begin{document}$${\chi(G)}$$\end{document} , is the minimum number of colors required in any proper coloring of G. A linear t-coloring of a graph is a proper t-coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths. The linear chromatic number of a graph G, denoted by \documentclass[12pt]{minimal}
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\begin{document}$${lc(G)}$$\end{document} , is the minimum t such that G has a linear t-coloring. In this paper, the linear t-colorings of Sierpiński-like graphs S(n, k), \documentclass[12pt]{minimal}
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\begin{document}$${S^+(n, k)}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${S^{++}(n, k)}$$\end{document} are studied. It is obtained that \documentclass[12pt]{minimal}
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\begin{document}$${lc(S(n, k))= \chi (S(n, k)) = k}$$\end{document} for any positive integers n and k, \documentclass[12pt]{minimal}
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\begin{document}$${lc(S^+(n, k)) = \chi(S^+(n, k)) = k}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${lc(S^{++}(n, k)) = \chi(S^{++}(n, k)) = k}$$\end{document} for any positive integers \documentclass[12pt]{minimal}
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\begin{document}$${n \geq 2}$$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$${k \geq 3}$$\end{document} . Furthermore, we have determined the number of paths and the length of each path in the subgraph induced by the union of any two color classes completely.