Local Antimagic Vertex Coloring of Gear Graph

Let G = (V, E) be a graph that consist of a vertex set V and an edge set E. The local antimagic labeling f of a graph G with edge-set E is a bijection map from E to {1, 2, …, |E|} such that w(u) ≠ w(v), where w(u) = ∑e ∈ E(u) f(e) and E(u) is the set of edges incident to u. In this labeling, every vertex v is assigned w(v) as its color. The minimum number of colors in a local antimagic labelling, is called a local antimagic chromatic number and denoted by χla (G). This paper contribution is to determine the local antimagic chromatic number χla (Gn) of a gear graph. A gear graph is a graph obtained by inserting additional vertex between each pair of adjacent vertices on the circumference of the wheel graph Wn. The gear graph Gn has 2n+1 vertices and 3n edges.