A new kind of graph coloring

Abstract

A proper k-coloring C 1 , C 2 , … , C k 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 C i . 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.