r-Dynamic Chromatic Number of Some Line Graphs

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

“…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. In particular, the last mentioned paper deals with the r-dynamic chromatic number of the line graph of a helm graph, which coincides as such with the so-called r-dynamic edge chromatic number of a helm graph.…”

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].…”

On dynamic colouring of cartesian product of complete graph with some graphs

“…Generally, a proper vertex k-colouring is called r-dynamic [5] if for every vertex v with degree at least r, the neighbors of v receive atleast r different colours. The smallest integer k such that G has a dynamic k-colouring is called r-dynamic chromatic number of G and denoted by χ r (G).…”