The Rainbow Saturation Number of Cycles

TL;DR

This work introduces and analyzes the rainbow saturation number $rsat(n,F)$ for edge-colored graphs, with a focus on cycles. It delivers an exact result $rsat(n,C_4)=3\left\lceil\frac{n-1}{2}\right\rceil$ for $n\ge5$ and establishes bounds for $C_5$ and general $C_r$, supported by explicit rainbow constructions (e.g., $M_n$, $W_n$, $\Omega_n$, Construction I/II) and a weight-transfer lower-bound framework. The authors also develop structural tools around bad and good roots to derive linear lower bounds and show that $rsat(n, C_r)$ can be bounded by $2n+O(r^2)$ for $r\ge7$, with tighter bounds in various regimes. Overall, the results demonstrate linear growth of $rsat(n, C_r)$ in $n$ with strategies that combine combinatorial constructions and path-based saturation arguments, contributing toward potential asymptotic characterizations in cycle rainbow saturation problems.

Abstract

An edge-coloring of a graph $H$ is a function $\mathcal{C}: E(H) \rightarrow \mathbb{N}$. We say that $H$ is rainbow if all edges of $H$ have different colors. Given a graph $F$, an edge-colored graph $G$ is $F$-rainbow saturated if $G$ does not contain a rainbow copy of $F$, but the addition of any nonedge with any color on it would create a rainbow copy of $F$. The rainbow saturation number $rsat(n,F)$ is the minimum number of edges in an $F$-rainbow saturated graph with order $n$. In this paper we proved several results on cycle rainbow saturation. For $n \geq 5$, we determined the exact value of $rsat(n,C_4)$. For $ n \geq 15$, we proved that $\frac{3}{2}n-\frac{5}{2} \leq rsat(n,C_{5}) \leq 2n-6$. For $r \geq 6$ and $n \geq r+3$, we showed that $ \frac{6}{5}n \leq rsat(n,C_r) \leq 2n+O(r^2)$. Moreover, we establish better lower bound on $C_r$-rainbow saturated graph $G$ while $G$ is rainbow.

Figures (7)

• Figure 1: The coloring of $M_n$.
• Figure 2: A rainbow graph $\mathcal{R}(W_n)$.
• Figure 3: An example of graph $\Omega_n$, where $\zeta_i=W_{n_i}$.
• Figure 4: A rainbow graph $\mathcal{R}(S_n)$. $B = K_a \cup K_b \cup tK_3$ where $t=\lfloor \frac{n-7}{3} \rfloor$ and $u$ is completely connected to $B$. $(a,b)=(3,3)$ if $n=3k+1$, $(a,b)=(4,3)$ if $n=3k+2$ and $(a,b)=(4,4)$ if $n=3k+3$ for some integer $k$.
• Figure 5: A rainbow graph $\mathcal{R}(\Gamma_n)$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Proposition 2.1
• Corollary 2.2