STM Article Repository

Let G = (V, E) be a graph and Cm be the cycle graph with m vertices. In this chapter, we continued Yeh’s work [1] on the distance labeling of the cycle graph Cm . An n-set distance labeling of a graph G is the labeling of the vertices (with n labels per vertex) of G under certain constraints determined by the distance between each pair of vertices in G. Following Yeh’s notation [1], the smallest value for the largest label in an n-set distance labeling of G is denoted by 1(n) (G) The basic findings were given in [1] for 1(2) (Cm) for all m and 1(n) (Cm) for some m where n 3. However, due to case-by-case difficulties, there were still gaps that remained unstudied. For these uncovered cases, we proved a lower bound for 1(n) (Cm). Then we developed an algorithm for finding an n-set distance labeling for n 3 based on our proof of the lower bound. We verified every single case for n = 3 up to n = 500 by this same algorithm, which revealed that the upper bound is the same as the lower bound for n 500. Potential future research application is briefly discussed at the end of this paper.