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graceful coloring; graceful chromatic numbers; tree
A proper coloring $c\colon V(G)\to \{1, 2,\ldots , k\}$, $k\ge 2$ of a graph $G$ is called a graceful $k$-coloring if the induced edge coloring $c'\colon E(G) \to \{1, 2, \ldots , k-1\}$ defined by $c'(uv)=|c(u)-c(v)|$ for each edge $uv$ of $G$ is also proper. The minimum integer $k$ for which $G$ has a graceful $k$-coloring is the graceful chromatic number $\chi _g(G)$. It is known that if $T$ is a tree with maximum degree $\Delta $, then $\chi _g(T) \le \lceil \frac 5{3}\Delta \rceil $ and this bound is best possible. It is shown for each integer $\Delta \ge 2$ that there is an infinite class of trees $T$ with maximum degree $\Delta $ such that $\chi _g(T)=\lceil \frac 5{3}\Delta \rceil $. In particular, we investigate for each integer $\Delta \ge 2$ a class of rooted trees $T_{\Delta , h}$ with maximum degree $\Delta $ and height $h$. The graceful chromatic number of $T_{\Delta , h}$ is determined for each integer $\Delta \ge 2$ when $1 \le h \le 4$. Furthermore, it is shown for each $\Delta \ge 2$ that $\lim _{h \to \infty } \chi _g(T_{\Delta , h}) = \lceil \frac 5{3}\Delta \rceil $.
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