Proper Rainbow Saturation Numbers for Cycles

TL;DR

This work investigates the proper rainbow saturation number $sat^*(n,F)$ for graphs under proper edge-colorings, focusing on rainbow-$F$-saturated graphs. It settles the asymptotics of $sat^*(n,C_4)$ by combining a constructive upper bound $sat^*(n,C_4)\lerac{11}{6}n+O(1)$ with a near-matching lower bound obtained via a dominating-set/decomposition analysis, and proves several structural lemmas about neighborhoods in rainbow-$C_4$-saturated graphs. Additionally, it provides upper bounds for longer cycles, giving $sat^*(n,C_5)\leloor{rac{5n}{2}}-4$ for $n\ge 9$ and $sat^*(n,C_6)\le rac{7}{3}n+O(1)$, via explicit core-dominating constructions and computer-assisted verification. The results illustrate that the proper rainbow saturation number can differ notably from ordinary saturation and highlight open questions for $C_k$ with larger $k$, including potential intervals of feasible color counts in such colorings. Overall, the paper advances the understanding of rainbow-saturated colorings and sets a path for future exploration of $sat^*(n,F)$ for broader families of graphs.

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 $\mathrm{sat}^*(n,F)$. 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 $\mathrm{sat}^*(n,C_6)$.

Figures (4)

• Figure 1: Subgraphs not appearing in $N(v)$
• Figure 6: A copy of $S_{1,2,2}$ in $N(u)$, colored to avoid a rainbow $C_4$-copy
• Figure 13: An edge-coloring of $G(n,C_5)$ containing no rainbow $C_5$-copy
• Figure 14: (a) An admissible edge-coloring of the core. (b) Non-core triangles and their connections to the core.

Theorems & Definitions (25)

• Theorem 1.1: Bushaw2022
• Theorem 1.2: KaszonyiTuza
• Theorem 1.3: Bushaw2022
• Theorem 1.4: Ollman
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7: Bushaw2022
• Theorem 1.8
• Theorem 1.9
• Proposition 2.1