Applying Skolem Sequences to Gracefully Label New Families of Triangular Windmills

TL;DR

This paper proves that Dutch windmills with three pendant triangles are (near) graceful, advancing the understanding of graph labelings and confirming Rosa's conjecture for this new family of triangular cacti.

Contribution

The paper introduces a novel application of Skolem sequences to establish graceful labelings for a new family of triangular windmill graphs.

Findings

Dutch windmills with three pendant triangles are (near) graceful.

Confirms Rosa's conjecture for this family of graphs.

Provides a new method using Skolem sequences for graph labeling.

Abstract

A function $f$ is a \textit{graceful labelling} of a graph $G=(V,E)$ with $m$ edges if $f$ is an injection $f:V\mapsto \{0,1,2,\dots,m\}$ such that each edge $uv \in E$ is assigned the label $|f(u)-f(v)|$, and no two edge labels are the same. If a graph G has a graceful labelling, we say that $G$ itself is graceful. In this paper, we prove any Dutch windmill with three pendant triangles is (near) graceful, which settles Rosa's conjecture for a new family of triangular cacti.