# A note on rainbow saturation number of paths

## TL;DR

This paper improves bounds on the rainbow saturation number of paths in edge-colored graphs, providing tighter upper bounds for certain path lengths and color counts.

## Contribution

It presents improved upper bounds for the rainbow saturation number of paths, refining previous results for specific path lengths and color parameters.

## Key findings

- Established new upper bounds for $sat_t(n,rak{R}(P_ ext{ell}))$
- Showed bounds are tighter for $ ext{ell} extgreater= 5$ and $t extgreater= 2 ext{ell} - 5$
- Enhanced understanding of rainbow saturation in colored graphs.

## Abstract

For a fixed graph $F$ and an integer $t$, the \dfn{rainbow saturation number} of $F$, denoted by $sat_t(n,\mathfrak{R}(F))$, is defined as the minimum number of edges in a $t$-edge-colored graph on $n$ vertices which does not contain a \dfn{rainbow copy} of $F$, i.e., a copy of $F$ all of whose edges receive a different color, but the addition of any missing edge in any color from $[t]$ creates such a rainbow copy. Barrus, Ferrara, Vardenbussche and Wenger prove that $sat_t(n,\mathfrak{R}(P_\ell))\ge n-1$ for $\ell\ge 4$ and $sat_t(n,\mathfrak{R}(P_\ell))\le \lceil \frac{n}{\ell-1} \rceil \cdot \binom{\ell-1}{2}$ for $t\ge \binom{\ell-1}{2}$, where $P_\ell$ is a path with $\ell$ edges. In this short note, we improve the upper bounds and show that $sat_t(n,\mathfrak{R}(P_\ell))\le \lceil \frac{n}{\ell} \rceil \cdot \left({{\ell-2}\choose {2}}+4\right)$ for $\ell\ge 5$ and $t\ge 2\ell-5$.