The Rainbow Saturation Number of Cycles

TL;DR

This paper investigates the minimum edges needed for rainbow-saturated graphs avoiding certain cycle subgraphs, providing exact and bounded values for cycles of length 4, 5, and r, with new bounds and results.

Contribution

It determines exact and improved bounds for rainbow saturation numbers of cycles, advancing understanding of rainbow saturation in edge-colored graphs.

Findings

Exact value of rsat(n,C4) for n ≥ 5

Bounds for rsat(n,C5) when n ≥ 15

Bounds for rsat(n,C_r) for r ≥ 6 and n ≥ r+3

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.