Graceful colorings of graphs with maximum degree three

A graceful $k$-coloring of a graph $G$ consists of a proper vertex coloring $f:V\left(G\right)\to \{1,\dots ,k\}$ that induces a proper edge coloring. In this case, the color assigned to an edge $uv\in E\left(G\right)$ is determined by the absolute value of the difference between the colors assigned to vertices $u$ and $v$. The minimum $k$ for which a graph $G$ has a graceful $k$-coloring is the graceful chromatic number of $G$, denoted by ${\chi }_g\left(G\right)$. In this paper, we establish the first known upper bound for the graceful chromatic number of an arbitrary graph, in terms of its maximum degree $\Delta$, i.e., we prove that the graceful chromatic number satisfies the inequality ${\chi }_g\left(G\right)\le \frac{5{\Delta }^2-3\Delta }{2}+1$. This result provides the first upper bound for ${\chi }_g\left(K_n\right)$ of complete graphs $K_n$, and represents an improvement over a previous finding by Bi et al. (2017) for regular complete $k$-partite graphs $K_{k\left(p\right)}$. We demonstrate the $\mathrm{NP}$-completeness of the problem of determining whether a graph $G$ has a graceful 5-coloring. Furthermore, we extend our investigation to subcubic graphs, establishing upper bounds for the graceful chromatic number of families of subcubic graphs without adjacent vertices of maximum degree. Additionally, we determine the graceful chromatic number for two cubic graph classes, namely Flower snarks and Goldberg snarks.