Starter Labelling of k-Windmill Graphs with Small Defects

Abstract

A graph on 2n vertices can be starter-labelled, if the vertices can be given labels from the nonzero elements of the additive group Z2n+1 such that each label i, either i or i-1, is assigned to exactly two vertices and the two vertices are separated by either i edges or i-1 edges, respectively. Mendelsohn and Shalaby have introduced Skolem-labelled graphs and determined the conditions of k-windmills to be Skolem-labelled. In this paper, we introduce starter-labelled graphs and obtain necessary and sufficient conditions for starter and minimum hooked starter labelling of all k-windmills.