D-local antimagic vertex coloring of a graph and some graph operations

Abstract

Let G(V,E) be a simple, connected (p,q)-graph. A d-local antimagic labelling is a bijection f: E(G) → {1,2,3,4,...q} such that for any two adjacent vertices, v₁ and v₂, w(v₁) ≠w(v₂) where w(v_i) = ∑e∈E(v_i) f(e) − deg(v_i),and E(v_i) is the set of edges incident to v_i for i = 1,2,..., p. Any d-local antimagic labelling induces a proper vertex coloring of G where the vertex, v_i is assigned the color w(v_i) for i = 1,2,... p and this coloring is called d-local antimagic coloring of G. The minimum number of colors required to color the vertices in a d-local antimagic coloring of G is called the d-local antimagic chromatic number of G and it is denoted as χ_dla(G). In this paper, we study the d-local antimagic vertex coloring of paths, cycles, star graphs, complete bipartite graphs and some graph operations such as the subdivision of each edge of a graph by a vertex and determine the exact value of the parameter, d-local antimagic chromatic number for these graphs.