Proper Rainbow Saturation Numbers for Cycles

We say that an edge-coloring of a graph G is proper if every pair of incident edges receive distinct colors, and is rainbow if no two edges of G receive the same color. Furthermore, given a fixed graph F, we say that G is rainbow F-saturated if G admits a proper edge-coloring which does not contain any rainbow subgraph isomorphic to F, but the addition of any edge to G makes such an edge-coloring impossible. The maximum number of edges in a rainbow F-saturated graph is the rainbow Tur\'an number, whose study was initiated in 2007 by Keevash, Mubayi, Sudakov, and Verstra\"ete. Recently, Bushaw, Johnston, and Rombach introduced study of a corresponding saturation problem, asking for the minimum number of edges in a rainbow F-saturated graph. We term this minimum the proper rainbow saturation number of F, denoted sat^*(n,F). We asymptotically determine sat^*(n,C₄), answering a question of Bushaw, Johnston, and Rombach. We also exhibit constructions which establish upper bounds for sat^*(n,C₅) and sat^*(n,C₆).