### 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, v

_{1}and v_{2}, w(v_{1}) ≠w(v_{2}) 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.**Published:**2022-05-02

**How to Cite this Article:**Preethi K. Pillai, J. Suresh Kumar, D-local antimagic vertex coloring of a graph and some graph operations, J. Math. Comput. Sci., 12 (2022), Article ID 149 Copyright © 2022 Preethi K. Pillai, J. Suresh Kumar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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