Some New Classes of (k, d) Graceful 3 Distance Trees and 3 Distance Unicyclic Graphs

Abstract

Objectives: To identify a new family of (k; d) graceful graphs. Methods : The methodology involves mathematical formulation for labeling of the vertices of a given graph and subsequently establishing that these formulations give rise to (k;d) graceful labeling. Findings: Here we define a three-distance tree as the tree possessing a path such that each vertex of the tree is at most at a distance three from that path. In this paper we identify two families of three distance trees that possess (k; d) graceful labeling. Furthermore, we show that the three distance unicyclic graphs obtained from these three distance trees by joining two end vertices of their central paths are also (k; d) graceful. Novelty: Here, we give (k; d) graceful labeling to two new families of graphs, namely some classes of three distance trees and three distance unicyclic graphs. This effort is the first of its kind which involves exploration of 3-distance (k; d) graceful graphs. Keywords: (k; D); graceful labelling; Hairy cycle; Firecracker; Three distance tree; Three distance unicyclic graphs