On a distance-constrained graph labeling to model cooperation

Inspired by distance-constrained graph labelings that model the assignment of frequencies to transmitters, we introduce a distance-constrained labeling of connected graphs that models the assignment of frequencies to oscillators. For a graph G and a vector of nonnegative real numbers k = (k(1), k(2), ..., k(p)), an L*(G; k)-labeling is any function f from V(G) into R such that for each i, 1 <= i <= p, vertical bar f(v) - f(w)vertical bar <= k(i) if d(v, w) = i. The span off is the absolute difference between the supremum and infimum of the image of f , and lambda(G,K)*, is the maximum span over all L*(G, k) labelings. In this paper, we explore the properties of lambda(G,K)* as a function of k, including continuity. And, for p = 2, we establish lambda(G,K)* when G is either a tree or a cycle. (C) 2021 Elsevier B.V. All rights reserved.