A new kind of graph coloring

A proper k-coloring C1,C2,…,Ck of a graph G is called strong if, for every vertex u∈V(G), there exists an index i∈{1,2,…,k} such that u is adjacent to every vertex of Ci. We consider classes SCOLOR(k) of strongly k-colorable graphs and show that the recognition problem of SCOLOR(k) is NP-complete for every k⩾4, but it is polynomial-time solvable for k=3. We give a characterization of SCOLOR(3) in terms of forbidden induced subgraphs. Finally, we solve the problem of uniqueness of a strong 3-coloring.