Backbone colorings along perfect matchings

Given a graph G = (V, E) and a spanning subgraph H of G (the backbone of G), a backbone coloring for G and H is a proper vertex coloring V → {1, 2, . . .} of G in which the colors assigned to adjacent vertices in H differ by at least two. In a recent paper, backbone colorings were introduced and studied in cases were the backbone is either a spanning tree or a spanning path. Here we study the case where the backbone is a perfect matching. We show that for perfect matching backbones of G the number of colors needed for a backbone coloring of G can roughly differ by a multiplicative factor of at most 4 3 from the chromatic number χ(G). We show that the computational complexity of the problem “Given a graph G with a perfect matching M , and an integer `, is there a backbone coloring for G and M with at most ` colors?” jumps from polynomial to NP-complete between ` = 3 and ` = 4. Finally, we consider the case where G is a planar graph.