Prime Vertex Labelings of Several Families of Graphs

Abstract

A simple and connected $n$-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 including cycle pendant stars, cycle chains, prisms, and generalized books.

Figures (24)

• Figure 1: The star $S_{4}$.
• Figure 2: Example of a unicyclic graph with five spurs.
• Figure 3: The graph $C_{5}\star P_{2}\star S_{6}$.
• Figure 4: The generalized prime vertex labeling of $C_{n}\star P_{2}\star S_{4}$.
• Figure 5: A prime vertex labeling of $C_{4}\star P_{2}\star S_{4}$.

Theorems & Definitions (17)

• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof