On r-dynamic Coloring of Ladder Graph and Tadpole Graph Using m-shadow Operation

Abstract

An r-dynamic coloring is a proper k-coloring of a graph G = {V,E} such that the neighbors of every vertex v ∈ V(G) is colored using ς: V(G) → S(c) where S(c) is a set of colors. The coloring is made in such a way that it satisfies the conditions: (i) For any edge uv ∈ E(G), the color of u and color of v are distinct and (ii) The cardinality of coloring the neighbors of any vertex v should be greater than or equal to min{r, d(v_G)}, where d(v_G) is the degree of the vertex v. In this paper the lower bounds for the r-dynamic coloring of m-shadow graph of ladder graph D_m(L_n) and tadpole graph D_m(L_n,p) are attained. Using the lower bounds the exact solution of r-dynamic chromatic number of the ladder graph L_n and tadpole graph L_n,p by m-shadow operation are obtained.