# Optimal proper connection of graphs

## TL;DR

This paper explores efficient methods to transform connected graphs into properly connected ones by recoloring edges, focusing on minimizing total recoloring and applying to specific graph classes like trees and bipartite graphs.

## Contribution

It introduces algorithms for converting monochromatic graphs into properly connected graphs with minimal recoloring, especially for trees, bipartite graphs, and graphs with independence number 2.

## Key findings

- Efficient recoloring algorithms for trees and bipartite graphs.
- Optimal recoloring strategies for graphs with independence number 2.
- Minimization of total recoloring and color changes in graph transformation.

## Abstract

An edge-colored graph $G$ is called properly colored if no two adjacent edges share a color in $G$. An edge-colored connected graph $G$ is called properly connected if between every pair of distinct vertices, there exists a path that is properly colored. In this paper, we discuss how to make a connected graph properly connected efficiently. More precisely, we consider the problem to convert a given monochromatic graph into properly connected by recoloring $p$ edges with $q$ colors so that $p+q$ is as small as possible. We discuss how this can be done efficiently for some restricted graphs, such as trees, complete bipartite graphs and graphs with independence number $2$.