[Paper Review] Minimum Coprime Labelings for Operations on Graphs

John Asplund, N. Bradley Fox|arXiv (Cornell University)|Jul 14, 2017

Graph Labeling and Dimension Problems5 references3 citations

TL;DR

This paper investigates the minimum coprime number, denoted 𝔪𝔠(G), for graphs that lack prime labelings—where adjacent vertices must have relatively prime labels using the smallest possible maximum label. It analyzes graph operations like unions, joins, coronas, and powers of paths and cycles, deriving exact formulas for 𝔪𝔠(G) in non-prime cases such as odd cycle unions, path and cycle joins, and coronas of complete graphs with small empty graphs, with key results showing 𝔪𝔠(Pₘ+Pₙ) = m+2n−2+(1−(−1)ᵐ)/2 and 𝔪𝔠(Cₘ+Cₙ) = m+2n−1−(−1)ᵐ under specific conditions.

ABSTRACT

A prime labeling of a graph of order $n$ is a labeling of the vertices with the integers $1$ to~$n$ in which adjacent vertices have relatively prime labels. A coprime labeling maintains the same criterion on adjacent vertices using any set of distinct positive integers. In this paper, we consider several families of graphs or products of graphs that have been shown to not have prime labelings and answer the natural question of how to label the vertices while minimizing the largest value in its set of labels.

Motivation & Objective

• To determine the minimum coprime number 𝔪𝔠(G) for graphs that do not admit prime labelings, extending beyond the standard {1,…,n} labeling.
• To analyze graph operations such as disjoint unions, joins, coronas, and graph powers (e.g., G², G³) on paths and cycles.
• To resolve open problems on minimum coprime labelings for non-prime graph families, including odd cycle unions, path-cycle joins, and coronas of complete graphs.
• To improve bounds on 𝔪𝔠(G) for trees and grid graphs, and to explore formulas for complete bipartite graphs Kₘ,ₙ and Kₙ,ₙ.
• To pose and investigate conjectures on the minimum coprime number for Kₙ⊙K̄ₘ and other non-prime graph families.

Proposed method

• Uses combinatorial labeling strategies based on prime and coprime sequences to assign distinct positive integers to vertices.
• Applies the principle that adjacent vertices must have relatively prime labels, minimizing the largest label used.
• Employs theorems on prime gaps and density (e.g., (14/13)⁹ < 2) to ensure label coprimality in constructed sequences.
• Constructs labelings by reserving odd integers and shifting prime labels to avoid conflicts in high-degree vertices.
• Leverages known results on primality of paths, cycles, and their operations to derive bounds and exact values.
• Uses symmetry and parity arguments (e.g., (−1)ᵐ) to handle cases based on the parity of graph orders in joins and unions.

Experimental results

Research questions

• RQ1 What is the minimum coprime number for the join of two paths, Pₘ+Pₙ, when it is not prime?
• RQ2 Can exact formulas be derived for the minimum coprime number of the join of two cycles, Cₘ+Cₙ, in non-prime cases?
• RQ3 What is the minimum coprime number for the corona Kₙ⊙K̄ₘ when prime labelings do not exist?
• RQ4 Can the minimum coprime number be determined for higher powers of paths and cycles (Gᵏ for k≥4)?
• RQ5 Is there a general formula for 𝔪𝔠(Kₘ,ₙ) when prime labelings are impossible, particularly for complete bipartite graphs?

Key findings

• For the join of two paths Pₘ+Pₙ with m≥n, the minimum coprime number is 𝔪𝔠(Pₘ+Pₙ) = m+2n−2+(1−(−1)ᵐ)/2, with exact values depending on the parity of m.
• For the join of two cycles Cₘ+Cₙ with m≥n and n≤10 or m>118 when n≥5, the minimum coprime number is 𝔪𝔠(Cₘ+Cₙ) = m+2n−1−(−1)ᵐ.
• The minimum coprime number for the join of a cycle and a path, Cₘ+Pₙ, is given by 𝔪𝔠(Cₘ+Pₙ) = m+2n−1−(−1)ᵐ when m≥n, and 𝔪𝔠(Cₘ+Pₙ) = n+2m−2+(1−(−1)ᵐ)/2 when n>m.
• For the corona Kₙ⊙K̄ₘ with m=1 or 2, the paper derives bounds and conjectures on 𝔪𝔠(Kₙ⊙K̄ₘ), with specific conjectures positing 𝔪𝔠(Kₙ⊙K̄ₘ) = pₙ₋₁ when mn≤pₙ₋₁−n−1 and 𝔪𝔠(Kₙ⊙K̄ₘ) = mn+n+1 otherwise.
• The paper confirms that the labeling strategy using odd integers and shifted primes yields a minimum coprime labeling for Pₘ when m≥10, with all odd labels below the largest odd prime used.
• Conjecture 28 posits that 𝔪𝔠(Pₘ+Pₙ) = m+2n−2+(1−(−1)ᵐ)/2 for all m≥n, suggesting a universal formula for path joins beyond the proven cases.