Totally Antimagic Total labeling of Ladders, Prisms and Generalised Pertersen graphs

TL;DR

This paper proves that ladders, prisms, and generalized Petersen graphs admit totally antimagic total labelings, and extends this property to chain graphs of such graphs, contributing to graph labeling theory.

Contribution

It establishes that specific classes of graphs, including ladders, prisms, and generalized Petersen graphs, are totally antimagic total graphs, and introduces the chain graph as also possessing this property.

Findings

Ladders are totally antimagic total graphs.

Prisms are totally antimagic total graphs.

Generalized Petersen graphs are totally antimagic total graphs.

Abstract

Given a graph $G$, a total labeling on $G$ is called edge-antimagic total (respectively, vertex-antimagic total) if all edge-weights (respectively, vertex-weights) are pairwise distinct. If a labeling on $G$ is simultaneously edge-antimagic total and vertex-antimagic total, it is called a totally antimagic total labeling. A graph that admits totally antimagic total labeling is called a totally antimagic total graph. In this paper, we prove that ladders, prisms and generalised Pertersen graphs are totally antimagic total graphs. We also show that the chain graph of totally antimagic total graphs is a totally antimagic total graph.