On r− dynamic coloring of the gear graph families

Abstract

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 r−dynamic chromatic number of the middle, central and line graphs of the gear graph.

The $r$-dynamic edge coloring of a closed helm graph

“…Both concepts generalize in a natural way those classical ones concerning the proper coloring and the chromatic number of a graph, which arise indeed when r = 1. Since the original work of Montgomery, the case r > 1 has widely been dealt with in the literature for different types of graphs [8,6,7,11,12,14,15,19,22,24], with particular emphasis in the case r = 2 [1,2,3,4,5]. Of particular interest for the topic of this paper, let us remark the studies of Mohanapriya et al [20] and Furmańczyk et al [9] concerning the r-dynamic coloring of different families of helm graphs.…”

On the r-dynamic coloring of some fan graph families

“…In the recent literature, one can find studies concerning different proper colorings of middle, total, central and line graphs of different types of graphs [5,27,28,30,31]. More specifically, the r-dynamic proper coloring of these four families of graphs has already been studied for bistar graphs [25], gear graphs [7] and helm graphs [10]. Some partial results concerning fan graphs F 1,n may be found in [23].…”