On the local edge (a, d)-Antimagic coloring of graphs: A new notion

Let G(V, E) be a connected, simple, and finite graph, with |V(G)| = p and |E(G)| = q. A bijection f : V (G) → {1,2,3,...,p} is called an edge antimagic labeling of graph if the element of the edge weight set w(uv) = f (u) + f (v), where uv ∈ E(G), are distinct. The edge antimagic labeling induces a local edge antimagic coloring of G if each edge e ∈ E(G) is colored by the weight w(e). The local edge antimagic coloring of graph is said to be a local edge (a, d)-antimagic coloring of G if the set of their edge colors form an arithmetic sequence with initial value a and different d. Furthermore, the local edge (a, d)-antimagic chromatic numbers, denoted by χle(a,d)(G), is the minimum number of colors needed to color G such that a graph G admits local edge (a, d)-antimagic coloring. In this paper, we will obtain the lower and upper bound of χle(a,d)(G) including to determine the exact of value of the local edge (a, d)-antimagic chromatic number of some graph classes.