Odd Prime Graph Labelings

TL;DR

This paper explores odd prime labelings in various graph classes, developing construction methods and advancing the conjecture that all prime graphs admit such labelings.

Contribution

It introduces new constructions for odd prime labelings across multiple graph classes and makes progress toward proving the conjecture for all prime graphs.

Findings

Constructed odd prime labelings for disjoint unions of cycles

Developed methods for stacked prisms and caterpillars

Progress toward proving all prime graphs have odd prime labelings

Abstract

An odd prime labeling is a variation of a prime labeling in which the vertices of a graph of order~$n$ are labeled with the distinct odd integers $1$ to $2n-1$ so that the labels of adjacent vertices are relatively prime. This paper investigates many different classes of graphs including disjoint unions of cycles, stacked prisms, and particular types of caterpillars, by using various methods to construct odd prime labelings. We also demonstrate progress toward proving a conjecture that all prime graphs have an odd prime labeling.