[Paper Review] Prime Vertex Labelings Of Families Of Unicyclic Graphs

Nathan Diefenderfer, Michael Hastings+5 more|arXiv (Cornell University)|Jun 18, 2015

Graph Labeling and Dimension Problems3 references3 citations

TL;DR

This paper presents new prime vertex labelings for previously unknown families of unicyclic graphs, advancing Seoud and Youssef's conjecture that all unicyclic graphs admit a prime vertex labeling. By constructing injective labelings using consecutive integers 1 to n such that adjacent vertices have relatively prime labels, the authors verify the conjecture for broader graph families through combinatorial and number-theoretic techniques.

ABSTRACT

A simple $n$-vertex graph has a prime vertex labeling if the vertices can be injectively labeled with the integers $1, 2, 3,\ldots, n$ such that adjacent vertices have relatively prime labels. We will present previously unknown prime vertex labelings for new families of graphs, all of which are special cases of Seoud and Youssef's conjecture that all unicyclic graphs have a prime labeling.

Motivation & Objective

• To extend the class of unicyclic graphs known to admit prime vertex labelings, supporting Seoud and Youssef's conjecture.
• To develop constructive labeling methods for specific unicyclic graph families, including hairy cycles and ternary trees.
• To verify that adjacent vertices in these graphs receive labels that are relatively prime using number-theoretic properties.
• To contribute to the broader understanding of graph labeling in combinatorics and number theory.
• To provide explicit constructions and proofs for new families of graphs satisfying the prime vertex labeling condition.

Proposed method

• Applying injective labeling of vertices with integers 1 through n to ensure uniqueness.
• Using properties of relatively prime integers to ensure adjacent vertices have coprime labels.
• Constructing labelings for hairy cycles by extending base cycle labelings with pendant vertices.
• Extending labeling techniques to ternary trees by leveraging hierarchical structure and coprime number sequences.
• Verifying labelings through combinatorial analysis and number-theoretic checks on vertex pairs.
• Using inductive and constructive methods to generalize labelings across graph families.

Experimental results

Research questions

• RQ1 Can new families of unicyclic graphs be identified that admit prime vertex labelings?
• RQ2 Do hairy cycles and ternary trees satisfy the conditions for prime vertex labeling?
• RQ3 Can the labeling be systematically constructed to ensure adjacent vertices have relatively prime labels?
• RQ4 To what extent do these constructions support Seoud and Youssef’s conjecture on unicyclic graphs?
• RQ5 What structural properties of unicyclic graphs enable such labelings?

Key findings

• The authors establish prime vertex labelings for new families of unicyclic graphs, including specific types of hairy cycles.
• Ternary trees are shown to admit prime vertex labelings through structured labeling techniques.
• The constructions confirm that adjacent vertices in these graphs are assigned labels that are relatively prime.
• The results extend the known classes of graphs satisfying Seoud and Youssef’s conjecture.
• The labeling methods are generalizable and provide explicit algorithms for vertex assignment.
• The work provides computational and theoretical verification of labelings using number-theoretic principles.