Folding = Colouring

TL;DR

This paper introduces the concept of graph foldings, showing that the folding number of any graph is equal to its chromatic number, linking folding operations to graph coloring.

Contribution

It establishes that the folding number of a graph is exactly its chromatic number, providing a new perspective on graph coloring through foldings.

Findings

Folding number equals chromatic number for all graphs

Folding operations can be used to determine graph colorability

Theoretical link between foldings and graph coloring

Abstract

The foldings of a connected graph $G$ are defined as follows. First, $G$ is a folding of itself. Let $G'$ be a graph obtained from $G$ by identifying two vertices at distance 2 in $G$. Then every folding of $G'$ is a folding of $G$. The folding number of $G$ is the minimum order of a complete folding of $G$. Theorem: The folding number of every graph equals its chromatic number.