On gap-labellings of some families of graphs

Abstract

This text addresses the proper gap-labelling problem in both its edge and vertex variants. Consider an assignment of integer values {1,…,k} to either the edges or vertices of a graph. In the edge-labelling variant, a vertex with degree at least two has its colour defined by the largest difference, or the largest “gap”, among the labels of its incident edges; vertices of degree 1 receive as induced colour the label assigned to its (only) incident edge. For the vertex variant, the gaps are defined by the largest difference among the labels in the vertices neighbourhood, and degree-one vertices receive as colour the label assigned to their (only) neighbour. A gap-labelling is proper if the obtained colouring is proper and the least number k of labels for which a graph admits a proper gap-labelling of its edges (vertices) is called the edge-gap (vertex-gap) number of the graph. In this article, we study both parameters, from a structural and algorithmic point of view, for traditional families of graphs such as Cycles, Crowns and Wheels. We also consider Unicyclic graphs and several families of Snarks.