Multi-Color Pebble Motion on Graphs

We consider a graph with n vertices, and p < n pebbles of m colors. A pebble move consists of transferring a pebble from its current host vertex to an adjacent unoccupied vertex. The problem is to move the pebbles to a given new color arrangement. We study the feasibility version of the problem-does a given instance have a solution? We use an algorithm of Auletta et al. (Algorithmica 23:223-245, 1999) for the problem where each pebble has a distinct color to give a linear time algorithm for the feasibility decision problem on a general graph.