Starter labelling of \(k\)-windmill graphs with small defects (Q499824)

Summary: A graph on \(2n\) vertices can be starter-labelled, if the vertices can be given labels from the nonzero elements of the additive group \(\mathbb Z_{2n+1}\) 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. \textit{E. Mendelsohn} and \textit{N. Shalaby} [Ars Comb. 53, 161--172 (1999; Zbl 0994.05133)] 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.