Graceful Cascading Labelling Algorithm: Construction of Graceful Labelling of Trees

The Graceful Labelling of trees is one of the most challenging conjectures in Graph Theory, proudly known as the 'Disease' of Graph Theory which remains a challenge as it remains unsolved. To counter the conjecture, an algorithm is proposed to construct a Graceful Binary Tree and a Graceful Caterpillar Tree. Here, the algorithm puts forward a solution to graceful labelling problem through a very efficient and simple approach. Most importantly, the Binary Tree exhibits the property of gracefulness and the construction of the tree remains one of the major contributions of the paper. The steps regarding the Algorithm are discussed and the other variants of the already known graceful graphs are discussed. Here, a basic initiative to prove the conjecture for complete binary trees as well as a proposed version of Binary Cascading Caterpillar Tree is put forward. The result is known that all graphs are graceful, here a different approach to construct a graceful graph is discussed. There are different works to prove special types of graphs that they have graceful labelling, but here we have tried to give an alternative approach. In this paper, we are proposing a simple method of graceful labelling Binary Cascading Caterpillar Trees and Complete Binary Tree.