Graceful labellings of variable windmills using Skolem sequences

TL;DR

This paper studies graceful labelings of windmill graphs using Skolem sequences, establishing existence results for various families and providing new characterizations of certain unions of cycles.

Contribution

It introduces methods to prove (near) graceful labelings of windmill graphs with specific cycle structures using Skolem-type sequences.

Findings

Graceful labelings exist for windmills with C3 and C4 vanes.

Infinite families of 3,5- and 3,6-windmills are (near) graceful.

Union of t 5-cycles is graceful iff t ≡ 0,3 mod 4, near graceful otherwise.

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 $t \equiv 0,3\!\!\pmod{4}$ and is near graceful when $t\equiv 1,2 \pmod{4}$.