Labelling Graphs with a Condition at Distance 2

Abstract

Given a simple graph G (V, E) and a positive number d, an Ld(2, 1)-labelling of G is a function f V(G) [0, oc) such that whenever x, y E V are adjacent, If(x)-f(Y)l >-2d, and whenever the distance between x and y is two, If(x) f(Y)l >d. The Ld(2, 1)-labelling number A(G, d) is the smallest number m such that G has an Ld(2, 1)-labelling f with max{f(v) v E V} m. It is shown that to determine A(G, d), it suffices to study the case when d 1 and the labelling is nonnegative integral-valued. Let A(G) A(G, 1). The labelling numbers of special classes of graphs, e.g., A(C) 4 for any cycle C, are described. It is shown that for graphs of maximum degree A, A(G) _ A + 2A. If G is diameter 2, A(G) _ A2, a sharp bound for some A. Determining A(G) is shown to be NP-complete by relating it to the problem of finding Hamilton paths.

Determining the L(2,1)-span in polynomial space

“…Griggs and Yeh [11] proved that λ(G) ≤ ∆ 2 + ∆ 1 and conjectured, that λ(G) ≤ ∆ 2 for every graph G. There are several results supporting this conjecture, for example Gonçalves [9] proved that λ(G) ≤ ∆ 2 + ∆ − 2 for graphs with ∆ ≥ 3. Havet et al [13] have settled the conjecture in affirmative for graphs with ∆ ≥ 10 69 .…”

“…According to [11], Roberts was the first who proposed a modification of this problem, which is called an L(2, 1)-labeling problem. It asks for such a labeling with nonnegative integer labels, that no vertices in distance 2 in a graph have the same label and labels of adjacent vertices differ by at least 2.…”

Solving L(2,1)-labeling Problem of Graphs using Genetic Algorithms

“…al. [6]. Note that However, determining λ(G) for general graphs has been proved to be NP-complete by Griggs et.…”

Distance Two labeling for Multi-Storey Graphs

“…Our work is motivated the conjecture of Griggs and Yeh [9] that asserts that λ 2,1 (G) ≤ ∆ 2 for every graph G with maximum degree ∆ ≥ 2. The conjecture has been verified only for several classes of graphs such as graphs of maximum degree two, chordal graphs [20] (see also [6,16]) and Hamiltonian cubic graphs [12,13].…”