A lower bound and several exact results on the d-lucky number

• Published in: Applied mathematics and computation Vol. 366; p. 124760
• Main Authors: Klavžar, Sandi, Rajasingh, Indra, Emilet, D. Ahima
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.02.2020
• Subjects: Cocktail-party graphs; Corona graphs; d-lucky labeling; Lucky labeling; Corona graphs; Cocktail-party graphs; 05C78; d-lucky labeling; Lucky labeling
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2019.124760

If ℓ:V(G)→N is a vertex labeling of a graph G=(V(G),E(G)), then the d-lucky sum of a vertex u ∈ V(G) is dℓ(u)=dG(u)+∑v∈N(u)ℓ(v). The labeling ℓ is a d-lucky labeling if dℓ(u) ≠ dℓ(v) for every uv ∈ E(G). The d-lucky number ηdl(G) of G is the least positive integer k such that G has a d-lucky labeling V(G) → [k]. A general lower bound on the d-lucky number of a graph in terms of its clique number and related degree invariants is proved. The bound is sharp as demonstrated with an infinite family of corona graphs. The d-lucky number is also determined for the so-called Gm,n-web graphs and graphs obtained by attaching the same number of pendant vertices to the vertices of a generalized cocktail-party graph.