On r-dynamic vertex coloring of some flower graph families

Abstract

Let [Formula: see text] be a simple, connected undirected graph with [Formula: see text] vertices and [Formula: see text] edges. Let vertex coloring [Formula: see text] of a graph [Formula: see text] be a mapping [Formula: see text], where [Formula: see text] and it is [Formula: see text]-colorable. Vertex coloring is proper if none of the any two neighboring vertices receives the similar color. An [Formula: see text]-dynamic coloring is a proper coloring such that [Formula: see text] min[Formula: see text], for each [Formula: see text][Formula: see text]. The [Formula: see text]-dynamic chromatic number of a graph [Formula: see text] is the minutest coloring [Formula: see text] of [Formula: see text] which is [Formula: see text]-dynamic k-colorable and denoted by [Formula: see text]. By a simple view, we exhibit that [Formula: see text], howbeit [Formula: see text] cannot be arbitrarily small. Thus, finding the result of [Formula: see text] is useful. This study gave the result of [Formula: see text]-dynamic chromatic number for the central graph, Line graph, Subdivision graph, Line of subdivision graph, Splitting graph and Mycielski graph of the Flower graph [Formula: see text] denoted by [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] respectively.

On the r-dynamic chromatic number of subdivision of wheel graph

“…It is obvious that χ r (S c1 (W n )) 5. we are able to verified that χ r (S c1 (W n )) = 5 for r=3, n=5.…”

“…it containing χ r (H) ≥ min {∆(H), r} + 1. chromatic numbers of special graph, such as the tadpol graph, the central of friendship graph, the rectangular book graph, and the triangular book graph, Gomathi et. al [5] constructed rdynamic chromatic numbers for four graphs (the central graph, Line graph, Splitting graph, and Subdivision graph of Subdivision graph as well as Mycielski graph of the Flower graph). The r-dynamic chromatic number has been studied by several authors, for example in [1,3,4,5,12].…”

“…al [5] constructed rdynamic chromatic numbers for four graphs (the central graph, Line graph, Splitting graph, and Subdivision graph of Subdivision graph as well as Mycielski graph of the Flower graph). The r-dynamic chromatic number has been studied by several authors, for example in [1,3,4,5,12].…”