International Journal of Combinatorics

Volume 2015, Article ID 528083, 5 pages

http://dx.doi.org/10.1155/2015/528083

## Starter Labelling of -Windmill Graphs with Small Defects

Department of Mathematics and Statistics, Memorial University of Newfoundland and Labrador, St. John’s, NL, Canada A1C 5S7

Received 25 March 2015; Accepted 9 July 2015

Academic Editor: Chris A. Rodger

Copyright © 2015 Farej Omer and Nabil Shalaby. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A graph on vertices can be starter-labelled, if the vertices can be given labels from the nonzero elements of the additive group such that each label , either or , is assigned to exactly two vertices and the two vertices are separated by either edges or edges, respectively. Mendelsohn and Shalaby have introduced Skolem-labelled graphs and determined the conditions of -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 -windmills.

#### 1. Introduction

Consider as an additive abelian group of odd order . A starter in is a partition of the nonzero elements of into unordered pairs such that . Starters were first used by Stanton and Mullin to construct Room squares [1]. Since then, starters have been widely used in several combinatorial designs such as Room cubes [2], Howell designs [3, 4], Kirkman triple systems [5], Kirkman squares and cubes [6, 7], Kotzig factorizations [8, 9], Hamilton path tournament designs [10], and optimal optical orthogonal codes [11]. A starter sequence of order is an integer sequence; of integers such that, for every , we consider either or such that or , respectively, and if with then . When is the additive inverse of in and if the inverse appears in the sequence, we call it a defect. For example, the sequence is a starter sequence of order 3 with one defect in the group . We notice that Skolem sequences are a special case of starter sequences when the number of defects is zero. It is well known that Skolem sequences and their generalizations have been used widely to construct several designs such as Room squares, one-factorizations, and round robin tournaments. In 1991, Mendelsohn and Shalaby [12] introduced the concept of Skolem labelling and also provided the necessary and sufficient conditions for Skolem labelling of paths and cycles. Eight years later, Mendelsohn and Shalaby [13] determined the condition for the existence of Skolem labelling for -windmills. In 2008, Baker and Manzer [14] obtained the necessary conditions for the Skolem labelling of generalized -windmills in which the vanes need not be of the same length and proved that these conditions are sufficient in the case where . In this paper, we introduce the concept of starter labelling of graphs and explore the necessary and the sufficient conditions for the existence of starter and minimum hooked starter labelling of -windmills. Furthermore, we restate the definitions of starter and hooked starter-labelled graphs.

*Definition 1. *A starter-labelled graph is a pair , where(a) is an undirected graph,(b),(c) exactly once for each ,(d)if and then violates .

*Definition 2. *A hooked starter-labelled graph is a pair satisfying the conditions of Definition 1 with instated of :.

*Example 3. *Figure 1 illustrates a hooked starter-labelled graph for 4-windmills.