Zero Ring Labeling of Graphs

On Efficient Zero Ring Labeling and Restricted Zero Ring Graphs

“…Although in some literature a zero ring is defined as the trivial ring that contains only one element, in this paper we follow the definition used by Acharya et al [3] and use the term zero ring for a ring with additive identity 0 such that ab = 0 for any a, b ∈ R. This notion of a zero ring was previously defined by Bourbaki in [4] and called it a pseudo ring of square zero.…”

“…It was shown that some families of trees have an efficient zero ring labeling. Although it was shown in [3] that every finite graph admits a zero ring labeling, not all graphs have an efficient zero ring labeling. In particular it was shown in [5] that some cycles do not have an efficient zero ring labeling.…”

“…In 2014, Acharya et al [3] introduced zero ring labeling. In this labeling, each vertex is assigned a unique label from a zero ring such that the sum of any two adjacent vertices is not zero, i.e., the additive identity of the zero ring.…”

“…The zero ring index of a graph, which is the smallest order of a zero ring in which the graph admits a zero ring labeling, was also studied for some well-known graphs. Acharya et al [2] determined a necessary and sufficient condition for a finite graph of order n to attain an optimal zero ring index.…”

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In [3], Acharya et al introduced the notion of a zero ring labeling of a connected graph G, where vertices are labeled by the elements of a zero ring such that the sum of the labels of adjacent vertices is not the additive identity of the ring. Archarya and Pranjali [1] also constructed a graph based on a finite zero ring called the zero ring graph.

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