Abstract
Coloring the vertices of a particular graph has often been motivated by its utility to various applied fields and its mathematical interest. A dynamic coloring of a graph G is a proper coloring of the vertex set V(G) such that for each vertex of degree at least 2, its neighbors receive at least two distinct colors. A dynamic k-coloring of a graph is a dynamic coloring with k colors. A dynamic k-coloring is also called a conditional (k, 2)-coloring. The smallest integer k such that G has a dynamic k-coloring is called the dynamic chromatic number chi _d(G) of G. In this paper, we investigate the dynamic chromatic number for the line graph of sunlet graph and middle graph, total graph and central graph of sunlet graphs, paths and cycles. Also, we find the dynamic chromatic number for Mycielskian of paths and cycles and the join graph of paths and cycles.
Highlights
Throughout this paper, all graphs are finite and simple
A dynamic coloring is defined as a proper coloring in which any multiple degree vertex is adjacent to more than one color class
For a regular graph G, it was shown by Alishahi: Theorem B [4] If G is an r -regular graph, χd (G) ≤ χ (G) + 14.06 log r + 1
Summary
Throughout this paper, all graphs are finite and simple. The dynamic chromatic number was first introduced by Montgomery [14]. For a regular graph G, it was shown by Alishahi: Theorem B [4] If G is an r -regular graph, χd (G) ≤ χ (G) + 14.06 log r + 1 Another upper bound on χd (G) is χd (G) ≤ 1 + l(G), where l(G) is the length of a longest path in G [14]. Alishahi [4] proved that for every graph G with χ (G) ≥ 4, χd (G) ≤ χ (G) + γ (G), where γ (G) is the domination number of a graph G Another upper bound for the dynamic chromatic number of a d-regular graph G in terms of χ (G) and the independence number of G, α(G), was introduced in [6].
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