Improper Graceful and Odd-graceful Labellings of Graph Theory

TL;DR

This paper introduces new graph labelling methods called in-improper and out-improper odd-graceful labellings, enabling the construction of larger graphs with these properties, including graphs with cycles.

Contribution

It proposes novel labelling schemes for trees that extend graceful labellings to graphs with cycles, broadening the scope of graph labelling techniques.

Findings

New in-improper and out-improper odd-graceful labellings for trees.

Construction of large graphs with these labellings.

Extension of graceful labellings to graphs with cycles.

Abstract

In this paper we define some new labellings for trees, called the in-improper and out-improper odd-graceful labellings such that some trees labelled with the new labellings can induce graceful graphs having at least a cycle. We, next, apply the new labellings to construct large scale of graphs having improper graceful/odd-graceful labellings or having graceful/odd-graceful labellings.