Local Distance Antimagic Labeling of Neighborhood Balanced Graphs

TL;DR

This work analyzes the local distance antimagic labeling and its induced coloring for neighborhood balanced colored graphs, investigating how common graph operations affect the local distance antimagic chromatic number $\chi_{ld}$. The authors provide explicit labelings and constructions that yield exact values (e.g., $\chi_{ld}(C_4^{(t)})=t+1$) and derive general bounds for graph products and lexicographic/composite structures. They establish both positive results (upper bounds like $\chi_{ld}(mG)\leq \chi_{ld}(G)$ under NBC, and $\chi_{ld}(G[H])\le 2\chi_{ld}(H)$ under suitable conditions)) and counterexamples illustrating when similarly simple bounds fail (e.g., Hedetniemi-type min bounds for direct products). Overall, the paper advances understanding of how local distance antimagic labelings interact with neighborhood-balanced colorings across a broad class of graph constructions, enriching the theory of graph labeling and its structural implications.

Abstract

Let G = (V, E) be a graph of order n without isolated vertices. A bijection f from vertex set of G to the set of integers from 1 to n is called a local distance antimagic labeling, if w(u) is not equal to w(v) for every edge uv of G, where w(u) is sum of labels of vertices adjacent to u. The local distance antimagic chromatic number xld(G) 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 article, we study the local distance antimagic labeling of neighborhood balanced colored graphs.

Figures (1)

• Figure 1: Neighborhood Balanced Coloring of $C_8$ and $K_{4,6}$.

Theorems & Definitions (60)

• Proposition 1.1
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