Geodesic
Dynamic proper vertex colorings of the graph
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part II, Tome 381 (2010), pp. 47-77
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Let a subdivision of the complete graph $K_n$ be any graph, which can be constructed from $K_n$ by substituting some edges of $K_n$ with chains of two edges (every such chain adds to a graph a new vertex of degree 2). Let $d\ge8$, $G$ is a connected graph with maximal vertex degree $d$. We proof that there is a proper dynamic vertex coloring of $G$ with $d$ colors iff $G$ is distinct from $K_{d+1}$ and its subdivisions. Bibl. 7 titles. 
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D. V. Karpov. Dynamic proper vertex colorings of the graph. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part II, Tome 381 (2010), pp. 47-77. http://geodesic.mathdoc.fr/item/ZNSL_2010_381_a2/

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