The edge-recoloring cost of monochromatic and properly edge-colored paths and cycles

Abstract

We introduce a number of problems regarding edge-color modifications in edge-colored graphs and digraphs. Consider a property π, a c-edge-colored graph Gc not satisfying π, and an edge-recoloring cost matrix R=[rij]c×c where rij≥0 denotes the cost of changing color i of edge e to color j. Basically, in this kind of problem the idea is to change the colors of one or more edges of Gc in order to construct a new edge-colored graph such that the total edge-recoloring cost is minimized and property π is satisfied. We also consider the destruction of potentially undesirable structures with the minimum edge-recoloring cost. In this paper, we are especially concerned with the construction and destruction of properly edge-colored and monochromatic paths, trails and cycles in graphs and digraphs. Some related problems and future directions are presented.