Some Results on Local Distance Antimagic Chromatic Number of Graphs

TL;DR

This paper investigates the local distance antimagic chromatic number of graphs, focusing on the join and lexicographic product of graphs with empty graphs, providing new insights into graph labelings and colorings.

Contribution

It introduces the study of local distance antimagic chromatic number for specific graph operations, expanding understanding of graph labelings and colorings.

Findings

Determined the local distance antimagic chromatic number for join graphs.

Analyzed the chromatic number for lexicographic products with empty graphs.

Provided bounds and exact values for specific graph classes.

Abstract

Let G=(V,E) be a graph of order n without isolated vertices. A bijection f:V -- {1,2,...n} is called a local distance antimagic labeling if the weights of any two adjacent vertices are not equal, where the weight of a vertex is defined to be the sum of labels of adjacent vertices. The local distance antimagic chromatic number is defined to be the minimum number of colors taken over all colorings of G induced by local distance antimagic labelings of G. In this paper, we study the local distance antimagic chromatic number for the join of graphs and the lexicographic product of graphs with empty graphs.