| Abstrakt: |
Abstract: A proper coloring c V(G)\to\{1, 2,\ldots, k\}, k\ge2 of a graph G is called a graceful k-coloring if the induced edge coloring c' 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\frac53\Delta\rceil and this bound is best possible. It is shown for each integer \Delta\ge2 that there is an infinite class of trees T with maximum degree \Delta such that \chi_g(T)=\lceil\frac53\Delta\rceil. In particular, we investigate for each integer \Delta\ge2 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\ge2 when 1 \le h \le4. Furthermore, it is shown for each \Delta\ge2 that \lim_{h \to\infty} \chi_g(T_{\Delta, h}) = \lceil\frac53\Delta\rceil. |
