k-Safe Labelings of Connected Graphs

In a k-safe labeling of a graph G, each vertex is labeled by a distinct positive integer such that the difference of the labels of two adjacent vertices is at least k. The span of a k-safe labeling of G is the range between the minimum and the maximum labels used in G. The k-safe labeling problem asks to label all the vertices of G using the minimum span. This problem has practical applications in assigning frequencies of transmitters in a network. k-safe labeling problem has been proven to be NP-hard and there is not an exact upper bound on the span of k-safe labeling of a graph. In this paper, we give an upper bound on k-safe labelings of all connected graphs based on the size of the maximum clique in the graph. Our proof leads to a polynomial-time algorithm for finding a k-safe labeling of any connected graph attaining the bound.