Coloring drawings of bipartite graphs: A problem in automated assembly

Abstract

Several workpiece carriers or vehicles are tethered to fixed bases. The vehicles are to visit a cycle of stations according to a given schedule. The vehicles can do this without tether collisions only if the bases and stations are feasibly located in the plane. Finding feasible configurations of bases and stations reduces, under idealizing assumptions, to a previously unexamined problem in graph coloring: find rectilinear planar drawings of the complete bipartite graph K m,n with m≤ n whose edges can be colored with n colors so that no two edges of the same color intersect. Useful but fragmentary results are reported: An exhaustive classification is found for feasible configurations of three or fewer bases and three or fewer stations; necessary and sufficient conditions for feasibility are found in the case of two vehicles and any number of stations; three classes of feasible configurations with unlimited numbers of bases and stations are identified, but whether these exhaust the possibilities is an open question. An answer to this question would be helpful to the designers of assembly systems using tethered workpiece carriers.