Bounds for the L(d, 1): number of diameter 2 graphs, trees and cacti Online publication date: Tue, 05-Sep-2006
by Anja Kohl
International Journal of Mobile Network Design and Innovation (IJMNDI), Vol. 1, No. 2, 2006
Abstract: Given a graph G = (V, E) and a positive integer d, an L(d, 1)-labelling of G is a function f : V (G) → {0, 1, …} such that if two vertices x and y are adjacent, then ' f (x) − f (y) ' ≥ d; if they are at distance 2, then ' f (x) − f (y) ' ≤ 1. The L(d, 1)-number of G, denoted by λd, 1(G), is the smallest number m such that G has an L(d, 1)-labelling with max{f (x) : x ∈ V (G)} = m. It is known that for diameter 2 graphs it holds λ2,1 ≤ Δ². We will prove Δ²+(d−2)Δ−1 as an upper bound for λd,1 for all diameter 2 graphs of order ≤ Δ² − 1 and d ≥ 2. For diameter 2 graphs of maximum degree Δ = 2, 3, 7 and order Δ² + 1 exact values or tight bounds for λd,1 will be presented. After this we provide a lower bound for λd,1 for trees. At last an upper bound for λd,1 for cacti will be established. In particular, we determine λ1,1 for all cacti and we will show λd,1 ≤ Δ + 2d − 2 for cacti having large girth or a large maximum degree Δ.
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