Graceful Coloring of Ladder Graphs

TL;DR

This paper investigates the graceful chromatic number of various ladder graph variants, providing new insights into their coloring properties and extending the understanding of graceful colorings in graph theory.

Contribution

It introduces new results on the graceful chromatic number for different variants of ladder graphs, expanding the existing knowledge in graph coloring theory.

Findings

Determined the graceful chromatic number for specific ladder graph variants.

Extended the theory of graceful colorings to new classes of ladder graphs.

Provided bounds and exact values for the graceful chromatic number in studied graphs.

Abstract

A graceful k-coloring of a non-empty graph $G=(V,E)$ is a proper vertex coloring $f:V(G)\rightarrow\lbrace 1,2,...,k \rbrace$, $k\geq 2$, which induces a proper edge coloring $f^{*}:E(G)\rightarrow\lbrace 1, 2, . . . , k-1 \rbrace $ defined by $f^{*}(uv) = |f(u)-f(v)|$, where $u,v\in V(G)$. The minimum $k$ for which $G$ has a graceful $k$-coloring is called graceful chromatic number, $\chi_{g}(G)$. The graceful chromatic number for a few variants of ladder graphs are investigated in this article.