An Analysis of Graceful Coloring in a Specific r-Regular Graphs

TL;DR

This paper investigates the graceful chromatic number of a specific class of regular graphs, focusing on complete graphs, using a set theoretic approach to address an open problem in graph coloring.

Contribution

It introduces a set theoretic method for computing the graceful chromatic number of complete graphs and characterizes graphs based on their graceful chromatic number.

Findings

Computed graceful chromatic number for complete graphs.

Provided characterizations of graphs based on their graceful chromatic number.

Addressed an open problem in regular graph coloring.

Abstract

A graceful $l$-coloring of a graph $G$ is a proper vertex coloring with $l$ colors which induces a proper edge coloring with at most $l-1$ colors, where the color for an edge $ab$ is the absolute difference between the colors assigned to the vertices $a$ and $b$. The graceful chromatic number $\chi_g(G)$ is the smallest $l$ for which $G$ permits graceful $l$-coloring. The problem of computing the graceful chromatic number of regular graphs is still open, though the existence of the lower bound was proved in \cite{3}. Hence, we pay attention to the computation of the graceful chromatic number of a special class of regular graphs namely complete graphs using set theoretic approach. Also, a few characterization of graphs based on their graceful chromatic number were examined.