Integer linear programming model and satisfiability test reduction for distance constrained labellings of graphs: the case of L (3,2,1)labelling for products of paths and cycles

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Let u and v be vertices of a graph G = (V, E) and d(u, v) be the distance between u and v in G. For positive integers k1, k2, …, kn with k1>k2>⋯>kn an L(k1, k2, …, kn)-labelling of G is a function f: V(G)→{0, 1, …} such that for every u, v ∈ V(G) and for all 1 ≤ i ≤ n, |f(u)− f(v)| ≥ ki if d(u, v) = i. The span of f is the difference between the largest and the smallest numbers in f(V(G)). The ,k2,…,kn-number of G is the minimum span over all L(k1, k2, …, kn)-labellings of G. In this study, an integer linear programming model and a satisfiability test reduction for an L(k1, k2, …, kn)-labelling are proposed. Both approaches are used for studying the λ3,2,1-numbers of strong, Cartesian and direct products of paths and cycles.