Distance Magic Labeling and Two Products of Graphs

Marcin Anholcer,Sylwia Cichacz,Aleksandra Tepeh,Iztok Peterin

doi:10.1007/s00373-014-1455-8

Abstract

Let $$G=(V,E)$$ be a graph of order $$n$$ . A distance magic labeling of $$G$$ is a bijection $$\ell :V\rightarrow \{1,\ldots ,n\}$$ for which there exists a positive integer $$k$$ such that $$\sum _{x\in N(v)}\ell (x)=k$$ for all $$v\in V $$ , where $$N(v)$$ is the neighborhood of $$v$$ . We introduce a natural subclass of distance magic graphs. For this class we show that it is closed for the direct product with regular graphs and closed as a second factor for lexicographic product with regular graphs. In addition, we characterize distance magic graphs among direct product of two cycles.

Highlights

Introduction and PreliminariesAll graphs considered in this paper are simple finite graphs
If a graph G admits a distance magic labeling, we say that G is a distance magic graph
In order to obtain a large class of graphs for which their direct product is distance magic we introduce a natural subclass of distance magic graphs

Summary

Introduction

Introduction and PreliminariesAll graphs considered in this paper are simple finite graphs. A distance magic graph G with an even number of vertices is called balanced if there exists a bijection : V (G) → {1, .

Results

Conclusion