1. [formula omitted]-labelings of subdivisions of graphs.
- Author
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Chang, Fei-Huang, Chia, Ma-Lian, Jiang, Shih-Ang, Kuo, David, and Liaw, Sheng-Chyang
- Subjects
- *
GRAPH labelings , *INTEGERS , *LABELS - Abstract
Given a graph G and a function h from E (G) to N , the h -subdivision of G , denoted by G (h) , is the graph obtained from G by replacing each edge u v in G with a path P : u x u v 1 x u v 2 ... x u v n − 1 v , where n = h (u v). When h (e) = c is a constant for all e ∈ E (G) , we use G (c) to replace G (h) . For a given graph G , an L (p , q) -labeling of G is a function f from the vertex set V (G) to the set of all nonnegative integers such that f (u) − f (v) ≥ p if d G (u , v) = 1 , and f (u) − f (v) ≥ q if d G (u , v) = 2. A k - L (p , q) -labeling is an L (p , q) -labeling such that no label is greater than k. The L (p , q) -labeling number of G , denoted by λ p , q (G) , is the smallest number k such that G has a k - L (p , q) -labeling. We study the L (p , q) -labeling numbers of subdivisions of graphs in this paper. We prove that λ p , q (G (3) ) = p + (Δ − 1) q when p ≥ 2 q and Δ > 2 p q , and show that λ p , q (G (h) ) = p + (Δ − 1) q when p ≥ 2 q and Δ ≥ 3 p q , where h is a function from E (G) to N so that h (e) ≥ 3 for all e ∈ E (G). [ABSTRACT FROM AUTHOR]
- Published
- 2021
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