For a graph G and a subgraph H (called backbone graph) of G, a backbone k-coloring of G with respect to H is a proper vertex coloring of G using colors from the set {1, 2, . . . , k}, with an additional condition that colors for any two adjacent vertices in H must differ by at least two. The backbone chromatic number of G over H, denoted by BBC(G,H), is the smallest k of a backbone k-coloring a...