  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.
