## Abstract

For an *r*-regular graph *G*, we define an edge-coloring *c* with colors from {1, 2, . . . , *k*}, in such a way that any vertex of *G* is incident with at least one edge of each color. The multiset-color *c*_{m}(*v*) of a vertex *v* is defined as the ordered tuple (*a*
_{1}, *a*
_{2}, . . . , *a*_{k}), where *a*_{i} (1 ≤ *i* ≤ *k*) denotes the number of edges of color *i* which are incident with *v* in *G*. Then this edge-coloring *c* is called a *k-kaleidoscopic coloring* of *G* if every two distinct vertices in *G* have different multiset-colors and in this way the graph *G* is defined as a *k-kaleidoscope*. In this paper, we determine the integer *k* for a complete graph *K*_{n} to be a *k*-kaleidoscope, and hence solve a conjecture in [P. Zhang, A Kaleidoscopic View of Graph Colorings, (Springer Briefs in Math., New York, 2016)] that for any integers *n* and *k* with *n* ≥ *k* + 3 ≥ 6, the complete graph *K*_{n} is a *k*-kaleidoscope. Then, we construct an *r*-regular 3-kaleidoscope of order $\left({}_{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}2}^{r-1}\right)-1$ − 1 for each integer *r* ≥ 7, where *r* ≡ 3 (mod 4), which solves another conjecture in [P. Zhang, A Kaleidoscopic View of Graph Colorings, (Springer Briefs in Math., New York, 2016)] on the maximum order of *r*-regular 3-kaleidoscopes.