# Questions on the Structure of Perfect Matchings inspired by Quantum Physics

## TL;DR

This paper explores the structure of perfect matchings in graphs, inspired by quantum physics, introducing a new concept called inherited vertex coloring to connect graph theory with quantum state constructability.

## Contribution

It introduces the concept of inherited vertex coloring and formulates related questions linking perfect matchings to quantum physics, bridging graph theory and photonic quantum technology.

## Key findings

- Formulated new graph-theoretical questions inspired by quantum physics.
- Introduced the concept of inherited vertex coloring for perfect matchings.
- Connected graph structures to quantum state constructability.

## Abstract

We state a number of related questions on the structure of perfect matchings. Those questions are inspired by and directly connected to Quantum Physics. In particular, they concern the constructability of general quantum states using modern photonic technology. For that we introduce a new concept, denoted as inherited vertex coloring. It is a vertex coloring for every perfect matching. The colors are inherited from the color of the incident edge for each perfect matching. First, we formulate the concepts and questions in pure graph-theoretical language, and finally we explain the physical context of every mathematical object that we use. Importantly, every progress towards answering these questions can directly be translated into new understanding in quantum physics.