Proper Rainbow Saturation Numbers for Cycles
<|reference_start|>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 $\mathrm{sat}^*(n,F)$. We asymptotically determine $\mathrm{sat}^*(n,C_4)$, answering a question of Bushaw, Johnston, and Rombach. We also exhibit constructions which establish upper bounds for $\mathrm{sat}^*(n,C_5)$ and $\mathrm{sat}^*(n,C_6)$.<|reference_end|>
@article{halfpap2024proper,
    title={Proper Rainbow Saturation Numbers for Cycles},
    author={Anastasia Halfpap, Bernard Lidick'y and Tom'av{s} Masav{r}'ik},
    journal={arXiv preprint arXiv:2403.15602},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.15602},
    primaryClass={math.CO cs.DM}
}