Coloring Graphs to Produce Properly Colored Walks

TL;DR

This paper investigates the minimum number of edge colors needed to ensure a properly colored walk exists between every pair of vertices in a graph, providing bounds for different graph classes and characterizations for bipartite graphs.

Contribution

It establishes upper bounds for the proper-walk connection number in cyclic and bridgeless graphs and characterizes bipartite graphs with a connection number of two.

Findings

Proper-walk connection number is at most three for all cyclic graphs.

Proper-walk connection number is at most two for bridgeless graphs.

Bipartite graphs with connection number two are characterized.

Abstract

For a connected graph, we define the proper-walk connection number as the minimum number of colors needed to color the edges of a graph so that there is a walk between every pair of vertices without two consecutive edges having the same color. We show that the proper-walk connection number is at most three for all cyclic graphs, and at most two for bridgeless graphs. We also characterize the bipartite graphs that have proper-walk connection number equal to two, and show that this characterization also holds for the analogous problem where one is restricted to properly colored paths.