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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