Skolem Number of Cycles and Grid Graphs

A Skolem sequence can be thought of as a labelled path where two vertices with the same label are that distance apart. This concept has naturally been generalized to labellings of other graphs, but always using at most two of any integer label. Given that more than two vertices can be mutually distance d apart, we define a new generalization of a Skolem sequence on graphs that we call a proper Skolem labelling. This brings rise to the question; “what is the smallest set of consecutive positive integers we can use to proper Skolem label a graph?” This will be known as the Skolem number of the graph. In this paper we give the Skolem number for cycles and grid graphs, while also providing other related results along the way.