# On Prime Labelings of Uniform Cycle Snake Graphs

### Abstract

A *prime labeling* of a graph of order *n* is an assignment of the integers 1, 2, ... , *n* to the vertices such that each pair of adjacent vertices has coprime labels. For positive integers *m, k, q* with *k* ≥ 3 and 1 ≤ q ≤ ⌊*k*/2⌋ , the *uniform cycle snake graph* *C ^{m}_{k,q}* is constructed by taking a path with

*m*edges and replacing each edge by a

*k*-cycle by identifying two vertices at distance

*q*in the cycle with the vertices of the original path edge. We construct prime labelings for

*C*for many pairs (

^{m}_{k,q}*k*,

*q*) and, in each case, all

*m*. These include: all cases with

*k*≤ 9 or

*k*= 11; all cases with

*q*= 2 when

*k*≡ 3 (mod 4); all cases with

*q*= 3 when

*k*has the form 2

^{a }

*−*1, 3

^{a}+ 1 or 3

^{b}+ 3 for all

*a*and all odd

*b*; all cases with

*q*= 4 when

*k*is even; and all cases with

*q*=

*k*/2 when

*q*is a prime congruent to 1 (mod 3).

*The PUMP Journal of Undergraduate Research*,

*6*, 151-171. Retrieved from https://journals.calstate.edu/pump/article/view/3633

Copyright (c) 2023 Agam Bedi, Matt Ollis, Samiksha Ramesh

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.