Subtractive Magic and Antimagic Total Labeling for Basic Families of Graphs

TL;DR

This paper investigates the existence of various subtractive magic and antimagic labelings in fundamental directed graph families, providing constructions and non-existence results.

Contribution

It introduces new subtractive labeling concepts and establishes their existence or non-existence for basic graph families, with explicit constructions.

Findings

Existence of subtractive arc-magic and antimagic labelings for certain graphs.

Non-existence results for some graph families.

Explicit labelings constructed where possible.

Abstract

A \textit{subtractive arc-magic labeling} (SAML) of a directed graph $G=(V,A)$ is a bijection $\lambda :V\cup A \to \{1,2,\ldots,|V|+|A|\}$ with the property that for every $xy\in A$ we have $\lambda(xy)+\lambda(y)-\lambda(x)$ equals to an integer constant. If $\lambda(xy)+\lambda(y)-\lambda(x)$ are distinct for every $xy\in A$, then $\lambda$ is a \textit{subtractive arc-antimagic labeling} (SAAL). A \textit{subtractive vertex-magic labeling} (SVML) of $G$ is such bijection with the property that for every $x\in V$ we have $\lambda(x)+\sum_{y\in V, yx \in A} \lambda(yx)-\sum_{y\in V, xy\in A} \lambda(xy)$ equals to an integer constant. If $\lambda(x)+\sum_{y\in V, yx \in A} \lambda(yx)-\sum_{y\in V, xy\in A} \lambda(xy)$ are distinct for every $x\in V$, then $\lambda$ is a \textit{subtractive vertex-antimagic labeling} (SVAL). In this paper we prove some existence or non-existence of SAML, SVML, SAAL, and SVAL for several basic families of directed graphs, such as paths, cycles, stars, wheels, tadpoles, friendship graphs, and general butterfly graphs. The constructions are given when such labeling(s) exists.