Further Results On k-Super Graceful Graphs

TL;DR

This paper investigates the properties of k-super graceful labelings in graphs composed of regular or bi-regular components, expanding understanding of graph labelings in these specific structures.

Contribution

It extends the study of k-super graceful labelings to graphs with regular or bi-regular components, providing new results and insights.

Findings

Identifies conditions under which such graphs are k-super graceful.

Provides classifications and examples of graphs with these labelings.

Establishes new theorems related to k-super gracefulness in regular structures.

Abstract

Let $G=(V(G),E(G))$ be a simple, finite and undirected graph of order $p$ and size $q$. For $k\ge 1$, a bijection $f: V(G)\cup E(G) \to \{k, k+1, k+2, \ldots, k+p+q-1\}$ such that $f(uv)= |f(u) - f(v)|$ for every edge $uv\in E(G)$ is said to be a $k$-super graceful labeling of $G$. We say $G$ is $k$-super graceful if it admits a $k$-super graceful labeling. In this paper, we study the $k$-super gracefulness of some graphs in which each component is either regular or bi-regular.