Ordered graceful labellings of the 2-star

Abstract

A variation on the well-known graceful labelling problem for graphs is that of finding an ordered graceful labelling. A graceful labelling f :V(T)→{1,2,…,n} of a tree T of order n is called ordered graceful if, when the edges of the tree are oriented from the endvertex with larger label to the endvertex with smaller label, then every vertex has either indegree 0 or outdegree 0. In this study, we consider a technique for obtaining ordered graceful labellings of a 2-star T of order n and central vertex r, motivated by a conjecture of Cahit (Bull. ICA 12 (1994) 15–18) that just two non-equivalent ordered graceful labellings f of T are possible when n=6 s+1 or 6 s+3 with f( r)=1, and just two are possible when n=6 s+3 or 6 s+5 with f( r)=2. We show this conjecture is false when s>1 by constructing 2 s non-equivalent ordered graceful labellings in each of these cases. We use a link with Skolem sequences to aid some of the constructions.