Prime and Odd Prime Labelings on Cycle-Related Graphs - Komarullah, Hafif; Hidayat, Noor; Krisnawati, Vira Hari; Wijaya, Kristiana (Research Center of Inorganic Materials and Coordination Complexes, FMIPA Universitas Sriwijaya, 2026)
Graph labeling is the process of determining integer values for vertices, edges, or both, based on certain criteria. Let G be a simple graph with the finite vertex set V(G). Prime labeling of G is a bijection ⍺:V(G)→1,2,…,|V(G)| for which each pair of adjacent vertices exhibits relatively prime labels. This concept has been extended to odd prime labeling, defined as a bijection ⍺:V(G)→ 1,3,...,2|V(G)|-1 satisfying the condition that the labels assigned to adjacent vertices are relatively prime labels. A graph that displays a (odd) prime labeling is designated as a (odd) prime graph. A recent conjecture state that every prime graph is an odd prime graph. In the present study, we conduct an investigation concerning prime and odd prime labeling, focusing on a range of cycle-related graphs classes. Our methods include the axiomatic descriptive approach and pattern detection techniques. We show that volcano graphs, C_3 ⨀_(x_1 y_0 ) F_n, C_3⊚K ̅_n, tadpole graphs, palm trees, and C_l ⨀_(x_1 y_0 ) mP_(n+1) are all both prime and odd prime graphs.