# Minimum Rainbow $H$-Decompositions of Graphs

## TL;DR

This paper investigates the decomposition of properly edge-colored graphs into rainbow copies of a fixed graph H and single edges, establishing a connection to classical minimum H-decompositions.

## Contribution

It introduces a new approach linking rainbow H-decompositions to traditional H-decompositions, expanding understanding of graph decompositions under edge colorings.

## Key findings

- Established a relation between rainbow and classical H-decompositions.
- Provided bounds and conditions for minimal rainbow H-decompositions.
- Extended the theory of graph decompositions to edge-colored graphs.

## Abstract

Given graphs $G$ and $H$, we consider the problem of decomposing a properly edge-colored graph $G$ into few parts consisting of rainbow copies of $H$ and single edges. We establish a close relation to the previously studied problem of minimum $H$-decompositions, where an edge coloring does not matter and one is merely interested in decomposing graphs into copies of $H$ and single edges.