Computation for Lucky Number of Some Families of Planar Graphs

Abstract

Let G be a finite simple graph and let λ(G) → 1, 2, . . . , k be a vertex labeling. For each vertex v ∈ V(G), define the induced neighborhood sum by Sλ(v) = ∑u∈N(v) λ(u). The labeling λ is called a lucky labeling if Sλ(u) ̸= Sλ(v) for every edge uv ∈ E(G), and the lucky number η(G) is the minimum value of k for which such a labeling exists. In this paper, we determine the lucky numbers of several structured families of planar graphs, including pentagonal, alternate pentagonal, and double pentagonal snake graphs; pentagonal book graphs; hairy cycle graphs; Jahangir graphs; and quadrilateral snake graphs with pendant edges together with their alternate forms. For each family, explicit vertex labelings are constructed and the corresponding neighborhood sums are evaluated to verify the lucky condition. The obtained lucky numbers are uniformly small, lying between 2 and 4, and within the considered families they depend at most on simple parity conditions rather than on the order of the graph. These constructions extend the catalogue of planar graphs with known exact lucky numbers and provide labeling patterns that may be adapted to related graph families.