Proper Rainbow Saturation Numbers for Cycles

Abstract

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 Turán number, whose study was initiated in 2007 by Keevash, Mubayi, Sudakov, and Verstraëte. 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 ${\mathrm{sat}}^{\ast }(n,F)$. We asymptotically determine ${\mathrm{sat}}^{\ast }(n,C_4)$, answering a question of Bushaw, Johnston, and Rombach. We also exhibit constructions which establish upper bounds for ${\mathrm{sat}}^{\ast }(n,C_5)$ and ${\mathrm{sat}}^{\ast }(n,C_6)$.

21 pages, 14 figures