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Distance Three Labelings of Trees∗
"... An L(2, 1, 1)labeling of a graph G assigns nonnegative integers to the vertices of G in such a way that labels of adjacent vertices differ by at least two, while vertices that are at distance at most three are assigned different labels. The maximum label used is called the span of the labeling, and ..."
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An L(2, 1, 1)labeling of a graph G assigns nonnegative integers to the vertices of G in such a way that labels of adjacent vertices differ by at least two, while vertices that are at distance at most three are assigned different labels. The maximum label used is called the span of the labeling, and the aim is to minimize this value. We show that the minimum span of an L(2, 1, 1)labeling of a tree can be bounded by a lower and an upper bound with difference one. Moreover, we show that deciding whether the minimum span attains the lower bound is an NPcomplete problem. This answers a known open problem, which was recently posed by King, Ras, and Zhou as well. We extend some of our results to general graphs and/or to more general distance constraints on the labeling. 1
Distance three labellings for direct products of three complete graphs
 Taiwanese J. Math
"... Abstract. The distance 3 labeling number λG(j0, j1, j2) for a graph G = (V, E) is the smallest integer α such that there is a function f: V → [0, α], satisfying f(u)−f(v)  ≥ jδ−1 for any pair of vertices u, v of distance δ ≤ 3. In this paper, we determine the distance 3 labeling number λG(j, k, 1 ..."
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Abstract. The distance 3 labeling number λG(j0, j1, j2) for a graph G = (V, E) is the smallest integer α such that there is a function f: V → [0, α], satisfying f(u)−f(v)  ≥ jδ−1 for any pair of vertices u, v of distance δ ≤ 3. In this paper, we determine the distance 3 labeling number λG(j, k, 1) for the direct product G = Kn×Km×K2 (n ≥ m ≥ 3) of 3 complete graphs under various conditions on j and k. As a consequence, we have the radio number rn(G) = 2mn − 1. 1.
Linear and cyclic distancethree labellings of trees
, 2013
"... Given a finite or infinite graph G and positive integers `, h1, h2, h3, an L(h1, h2, h3)labelling of G with span ` is a mapping f: V (G) → {0, 1, 2,..., `} such that, for i = 1, 2, 3 and any u, v ∈ V (G) at distance i in G, f(u)−f(v)  ≥ hi. A C(h1, h2, h3)labelling of G with span ` is defined ..."
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Given a finite or infinite graph G and positive integers `, h1, h2, h3, an L(h1, h2, h3)labelling of G with span ` is a mapping f: V (G) → {0, 1, 2,..., `} such that, for i = 1, 2, 3 and any u, v ∈ V (G) at distance i in G, f(u)−f(v)  ≥ hi. A C(h1, h2, h3)labelling of G with span ` is defined similarly by requiring f(u) − f(v) ` ≥ hi instead, where x ` = min{x, `− x}. The minimum span of an L(h1, h2, h3)labelling, or a C(h1, h2, h3)labelling, of G is denoted by λh1,h2,h3(G), or σh1,h2,h3(G), respectively. Two related invariants, λ h1,h2,h3