On $k$-Super Graceful Labeling of Graphs

TL;DR

This paper investigates the properties and conditions for $k$-super graceful labelings in graphs, providing new sufficient and necessary conditions for various bipartite and tripartite graph families.

Contribution

It introduces new criteria for $k$-super gracefulness and explores its application to standard bipartite and tripartite graphs, expanding understanding of graph labelings.

Findings

Established sufficient conditions for $k$-super gracefulness.

Identified necessary conditions for certain graph families.

Analyzed properties of $k$-super graceful labelings in standard graphs.

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 standard graphs. Some general properties are obtained. Particularly, we found many sufficient conditions on $k$-super gracefulness for many families of (complete) bipartite and tripartite graphs. We show that some of the conditions are also necessary.