In this paper, we determine the r-dynamic chromatic number of the fan graph Fm,n
and determine sharp bounds of this graph invariant for four related families of graphs: The middle graph M(Fm,n
), the total graph T (Fm,n
), the central graph C(Fm,n
) and the line graph L(Fm,n
). In addition, we determine the r-dynamic chromatic number of each one of these four families of graphs in case of being m = 1.
An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that jc(N (v))j min fr; d(v)g, for each v 2 V (G). The rdynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of middle, total, central and line graph of Bistar graph.
An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, d(v)}, for each v ∈ V(G). The r-dynamic chromatic number of a graph G is the minimum k such that G has an r-dynamic coloring with k colors. In this paper, we obtain the δ-dynamic chromatic number of the central vertex join of two graphs.
In this paper, we obtain explicitly the [Formula: see text]-dynamic chromatic number of the direct product of a path [Formula: see text] with a [Formula: see text]-subdivision [Formula: see text] of a star graph, for every positive integers [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Particularly, it is obtained that [Formula: see text].
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