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3-dynamic coloring of planar triangulations
Asayama, Yoshihiro,Kawasaki, Yuki,Kim, Seog-Jin,Nakamoto, Atsuhiro,Ozeki, Kenta North-Holland Pub. Co 2018 Discrete mathematics Vol.341 No.11
<P><B>Abstract</B></P> <P>An r -dynamic k -coloring of a graph G is a proper k -coloring such that any vertex v has at least min { r , <SUB> deg G </SUB> ( v ) } distinct colors in <SUB> N G </SUB> ( v ) . The r <I>-dynamic chromatic number</I> χ r d ( G ) of a graph G is the least k such that there exists an r -dynamic k -coloring of G .</P> <P>Loeb et al. (2018) showed that if G is a planar graph, then χ 3 d ( G ) ≤ 10 , and there is a planar graph G with χ 3 d ( G ) = 7 . Thus, finding an optimal upper bound on χ 3 d ( G ) for a planar graph G is a natural interesting problem. In this paper, we show that χ 3 d ( G ) ≤ 5 if G is a planar triangulation. The upper bound is sharp.</P>