Zero Ring Index of Cactus Graphs
M. Dela Rosa-Reynera and L. Aquino-Ruivivar (pp. 37-43)
Abstract
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 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 have also zero ring indices equal to their orders. The labeling was done using the zero ring M02(Zn).