Geodesic
Strong $k$-colorings of graphs
Diskretnaya Matematika, Tome 13 (2001) no. 1, pp. 78-89

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We say that a $k$-colouring $C_1,\ldots,C_k$ of a graph $G$ is strong if for any vertex $u\in VG$ there exists an index $i\in\{1,\ldots,k\}$ such that $u$ is adjacent to any vertex of the class $C_i$. We consider the class $S(k)$ of strongly $k$-colourable graphs and demonstrate that the problem to recognise $S(k)$ is NP-complete for any $k\ge 4$, whereas it is polynomially solvable for $k=3$. We characterise the class $S(3)$ in terms of forbidden induced subgraphs and solve the problem of uniqueness of a strong 3-colouring. 
@article{DM_2001_13_1_a4,
     author = {I. \'E. Zverovich},
     title = {Strong $k$-colorings of graphs},
     journal = {Diskretnaya Matematika},
     pages = {78--89},
     publisher = {mathdoc},
     volume = {13},
     number = {1},
     year = {2001},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DM_2001_13_1_a4/}
}
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I. É. Zverovich. Strong $k$-colorings of graphs. Diskretnaya Matematika, Tome 13 (2001) no. 1, pp. 78-89. http://geodesic.mathdoc.fr/item/DM_2001_13_1_a4/