On local distance antimagic labeling of graphs

Abstract

Let (Formula presented.) be a graph of order n and let (Formula presented.) be a bijection. For every vertex (Formula presented.), we define the weight of the vertex v as (Formula presented.) where N(v) is the open neighborhood of the vertex v. The bijection f is said to be a local distance antimagic labeling of G if (Formula presented.) for every pair of adjacent vertices (Formula presented.). The local distance antimagic labeling f defines a proper vertex coloring of the graph G, where the vertex v is assigned the color w(v). We define the local distance antimagic chromatic number (Formula presented.) to be the minimum number of colors taken over all colorings induced by local distance antimagic labelings of G. In this paper we obtain the local distance antimagic labelings for several families of graphs including the path Pn, the cycle Cn, the wheel graph Wn, friendship graph Fn, the corona product of graphs (Formula presented.), complete multipartite graph and some special types of the caterpillars. We also find upper bounds for the local distance antimagic chromatic number for these families of graphs.