A new notion of graph labeling called zero ring labeling is realized by assigning distinct elements of a zero ring to the vertices of the graph such that the sun of the labels of adjacent vertices is not equal to the additive identity of the zero ring. The zero ring index of a graph G is the smallest positive integer ξ (G) such that there exists a zero ring of order ξ (G) for which G admits a zero ring labeling. Any zero ring labeling of G is optimal if it uses a zero ring consisting of ξ(G) elements. It is known that any free of order n has a zero ring index equal to n. Considering that cactus graphs are an interesting generalization of trees, in this paper, we extend the optimal zero ring labeling scheme for trees to cactus graphs that leads us to establish that cactus graphs also have zero ring indices equal to their orders. The labeling was done using the zero ring M02(Zn).
Keywords: zero ring, zero ring labeling, zero ring index, cactus graph, spanning treeAcharya, M., Pranjali, & Gupta, P. (2015). Zero ring labeling of graphs. Electronic NNotes in DDiscrete MMathematics, 48, 65-–72. Retrieved from https://www.sciencedirect.com/science/article/pii/S1571065315000116
Chartrand, G., & Zhang, P. (2005). Introduction to Graph Theory. Singapore: McGraw Hill.
Pranjali & Acharya, M. (2014). Graphs associated with finite zero rings. General Mathematics Notes, 24(2), 53-–69. Retrieved from https://www.emis.de/journals/GMN/yahoo_site_admin/assets/docs/6_GMN-6252-V24N2.334222050.pdf
Pranjali, Acharya, M., & Gupta, P. (2014). Further results on zero ring labeling of graphs. Bulletin of the International Mathematical Virtual Institute, 5, 205-–210. Retrieved from http://www.imvibl.org/buletin/bulletin_imvi_5_2_2015/bulletin_imvi_5_1_2015_205_210.pdf
Reynera, M. D., & Ruivivar, L. A. (2017). Optimal zero ring labeling scheme for trees. Matimyas Matematika, 40(1), 31-–39. Retrieved from http://mathsociety.ph/matimyas/images/vol40/Reynera-Matimyas-v2.pdf
Reynera, M. D., & Ruivivar, L. A. (2018). On some families of graphs having zero ring indices equal to their orders. DLSU Research Congress Proceedings.
Pranjali & Acharya, M. (2014). Graphs associated with finite zero rings. General Mathematics Notes, 24(2), 53-–69. Retrieved from https://www.emis.de/journals/GMN/yahoo_site_admin/assets/docs/6_GMN-6252-V24N2.334222050.pdf
Pranjali, Acharya, M., & Gupta, P. (2014). Further results on zero ring labeling