pp 37-43 (Vol 12 2019)

M. Dela Rosa-Reynera 1*, and
L. Aquino-Ruivivar

1 Mathematics Department, Mariano Marcos State University, Quiling Sur, Batac City, Philippines
2 Mathematics and Statistics Department, De La Salle University, Manila, Philippines

Corresponding Author: reyneramichelle@gmail.com; michelle_reynera@dlsu.edu.ph


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 sum 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 tree of order n has a zero ring index equal to n. Considering that cactus graphs are interesting generalizations 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 have also zero ring indices equal to their orders.