On edge colorings of graphs with no color-rich cycles

Motivated by the broad studies of various edge-coloring anti-Ramsey-like problems where particular subgraphs are forbidden to be colored in a rainbow way or using many colors in general, we investigate edge-colorings with the constraint that each cycle sees at most $i$ colors, for a given positive integer $i$. For such a coloring, the goal is to find the maximum number of colors, denoted by $K_i^{\circ }\left(G\right)$. We explore the basic properties and estimates of $K_i^{\circ }\left(G\right)$. A greedy-based general lower bound on this invariant is provided and its sharpness is discussed; we show that, for small values of $i$, this bound is attained for highly connected graphs.