Abstract
A proper t-coloring of a graph G is a mapping \({\varphi: V(G) \rightarrow [1, t]}\) such that \({\varphi(u) \neq \varphi(v)}\) if u and v are adjacent vertices, where t is a positive integer. The chromatic number of a graph G, denoted by \({\chi(G)}\) , 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 \({lc(G)}\) , 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), \({S^+(n, k)}\) and \({S^{++}(n, k)}\) are studied. It is obtained that \({lc(S(n, k))= \chi (S(n, k)) = k}\) for any positive integers n and k, \({lc(S^+(n, k)) = \chi(S^+(n, k)) = k}\) and \({lc(S^{++}(n, k)) = \chi(S^{++}(n, k)) = k}\) for any positive integers \({n \geq 2}\) and \({k \geq 3}\) . 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.
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Xue, B., Zuo, L., Wang, G. et al. The Linear t-Colorings of Sierpiński-Like Graphs. Graphs and Combinatorics 30, 755–767 (2014). https://doi.org/10.1007/s00373-013-1289-9
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DOI: https://doi.org/10.1007/s00373-013-1289-9