𝒌 – Graceful Labeling of Triangular Type Grid Graphs 𝑫𝒏(𝑷𝒎) and 𝑳 – Vertex Union of 𝑫𝒏(𝑷𝒎)

Abstract

Graph labeling is one of the most popular research topics in the field of graph theory. Prime labeling, antimagic labeling, radio labeling, graceful labeling, lucky labeling, and incidence labeling are some of the labeling techniques. Among the above-mentioned techniques, graceful labeling is one of the most engaging graph labeling techniques with a vast amount of real-world applications. Over the past few decades, plenty of studies have been conducted on this area in various dimensions. Grid graphs are very much useful in applications of circuit theory, communication networks, and transportation networks. However, in the literature, there are not many research papers on the graceful labeling of grid graphs except a few on odd graceful labeling. In our work, we prove that triangular-type grid graphs, 𝐷𝑛(𝑃𝑚) and 𝐿 – vertex union of 𝐷𝑛(𝑃𝑚) admit 𝑘 – general graceful labeling and 𝑘 – even and 𝑘 – odd graceful labeling. Further, we introduce combinatorial proofs for them as well.