Bounds on Graceful k-colorings of graphs

Abstract

We investigate the graceful $k$-coloring introduced by Gary Chartrand in 2015. The graceful $k$-coloring of a graph $G$ consists of a proper vertex coloring $\pi :V\left(G\right)\to \{1,2,\dots ,k\}$, $k\ge 1$, that induces a proper edge coloring ${\pi }^{\prime }:E\left(G\right)\to \{1,2,\dots ,k-1\}$ defined by ${\pi }^{\prime }\left(uv\right)=|\pi \left(u\right)-\pi \left(v\right)|$, where $uv\in E\left(G\right)$. The smallest positive integer $k$ for which a graph $G$ has a graceful $k$-coloring is called the graceful chromatic number of $G$ and is denoted by ${\chi }_g\left(G\right)$. In this paper, we improve a previous upper bound for the graceful chromatic number of an arbitrary graph, showing that ${\chi }_g\left(G\right)\le 2{\Delta }^2-\Delta +1$, where $\Delta$ is the maximum degree of the graph $G$. This also implies an improvement over the first bound given for complete graphs. Moreover, we study this problem within the context of cubic graph classes, determining the exact value of the graceful chromatic number of each member of the infinite families of the Generalized Blanuša and the Loupekine snarks.