Graceful labellings of variable windmills using Skolem sequences

Abstract

In this paper, we introduce graceful and near graceful labellings of several families of windmills. In particular, we use Skolem-type sequences to prove (near) graceful labellings exist for windmills with $C_3$ and $C_4$ vanes, and infinite families of $3,5$-windmills and $3,6$-windmills. Furthermore, we offer a new solution showing that the graph obtained from the union of $t$ 5-cycles with one vertex in common ($C_5^t$) is graceful if and only if $t\equiv 0,3(\mathrm{mod}4)$ and is near graceful when $t\equiv 1,2(\mathrm{mod}4)$.

38 pages (including 8 pages of appendices), 1 figure