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A proper coloring $c:V\left(G\right)\to \{1,2,...,k\}$, $k\ge 2$ of a graph $G$ is called a graceful $k$-coloring if the induced edge coloring $c^{\text{'}}:E\left(G\right)\to \{1,2,...,k-1\}$ defined by $c^{\text{'}}\left(uv\right)=|c\left(u\right)-c\left(v\right)|$ 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 ${\chi }_g\left(G\right)$. It is known that if $T$ is a tree with maximum degree $\Delta$, then ${\chi }_g\left(T\right)\le \lceil \frac{5}{3}\Delta \rceil$ 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 ${\chi }_g\left(T\right)=\lceil \frac{5}{3}\Delta \rceil$. In particular, we investigate for each integer $\Delta \ge 2$ a...

For an r-regular graph G, let c : E(G) → [k] = 1, 2, . . . , k, k ≥ 3, be an edge coloring of G, where every vertex of G is incident with at least one edge of each color. For a vertex v of G, the multiset-color cm(v) of v is defined as the ordered k-tuple (a1, a2, . . . , ak) or a1a2 … ak, where ai (1 ≤ i ≤ k) is the number of edges in G colored i that are incident with v. The edge coloring c is called k-kaleidoscopic if cm(u) ≠ cm(v) for every two distinct vertices u and v of G. A regular graph...