Graceful Labeling of the Upstairs Graph

Andika Ellena Saufika Hakim Maharani, Mamika Ujianita Romdhini, Abdurahim, Lingga Gita Dwikasari, Natasya Dyahayu Lestari

Abstract

Determining whether an arbitrary graph admits a graceful labeling remains an open problem in graph theory, as no general characterization of graceful graphs has been established. This study introduces the Upstairs graph, denoted by (Us_n), as a new graph structure based on a layered construction of grid graphs. Its graceful labeling properties are investigated through theoretical analysis and computational verification using a Python algorithm that checks the labeling conditions and generates visualizations of labeled graphs. The study provides a formal definition of the Upstairs graph and presents a general theorem stating that (Us_n) admits a graceful labeling for every integer (n\geq1). Illustrative cases demonstrate the labeling construction, while computational results are consistent with the theoretical results for the tested instances. Thus, the Upstairs graphs constitute a new family of grid-derived graphs with graceful labeling properties.

 

Keywords: graceful labeling; graph construction; graph labeling; grid graph; Python algorithm; Upstairs graph.

 

DOI https://doi.org/10.55463/issn.1674-2974.53.8.11


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