Proper-walk connection number of graphs

TL;DR

This paper investigates the proper-walk connection number in graphs, establishing polynomial-time solvability and characterizing graphs based on the minimum number of colours needed for proper connectivity.

Contribution

It proves the problem can be solved in polynomial time and provides a complete characterization of graphs by their proper-walk connection number.

Findings

Polynomial-time algorithm for proper-walk connection number

Complete characterization of graphs by their connection number

Solved an open problem in graph connectivity theory

Abstract

This paper studies the problem of proper-walk connection number: given an undirected connected graph, our aim is to colour its edges with as few colours as possible so that there exists a properly coloured walk between every pair of vertices of the graph i.e. a walk that does not use consecutively two edges of the same colour. The problem was already solved on several classes of graphs but still open in the general case. We establish that the problem can always be solved in polynomial time in the size of the graph and we provide a characterization of the graphs that can be properly connected with $k$ colours for every possible value of $k$.