Kaleidoscopic Colorings of Graphs

For an -regular graph , let : ) → [ ] = {1, 2, . . . , }, ≥ 3, be an edge coloring of , where every vertex of is incident with at least one edge of each color. For a vertex of , the multiset-color ) of is defined as the ordered -tuple ( , , . . . , ) or … , where (1 ≤ ≤ ) is the number of edges in colored that are incident with . The edge coloring is called -kaleidoscopic if ) ≠ ) for every two distinct vertices and of . A regular graph is called a -kaleidoscope if has a -kaleidoscopic coloring. It is shown that for each integer ≥ 3, the complete graph is a -kaleidoscope and the complete graph is a 3-kaleidoscope for each integer ≥ 6. The largest order of an -regular 3-kaleidoscope is . It is shown that for each integer ≥ 5 such that ≢ 3 (mod 4), there exists an -regular 3-kaleidoscope of order .