N-Set Distance-Labelings for Cycle Graphs

Let G = (V, E) be a graph and C_m be the cycle graph with m vertices. In this paper, we continued Yeh’s work [1] on the distance labeling of the cycle graph C_m. 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 depending on the distance between each pair of the 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 λ₁⁽ⁿ⁾(G). Basic results were presented in [1] for λ₁⁽²⁾(C_m) for all m and λ₁⁽ⁿ⁾(C_m) for some m where n ≥ 3. However, there were still gaps left unstudied due to case-by-case complexities. For these uncovered cases, we proved a lower bound for λ