Geodesic
Minimal rankings of the Cartesian product Kₙ ☐ Kₘ
Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 725-735

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For a graph G = (V, E), a function f:V(G) → 1,2, ...,k is a k-ranking if f(u) = f(v) implies that every u - v path contains a vertex w such that f(w) > f(u). A k-ranking is minimal if decreasing any label violates the definition of ranking. The arank number, ψ_r(G), of G is the maximum value of k such that G has a minimal k-ranking. We completely determine the arank number of the Cartesian product Kₙ ☐ Kₙ, and we investigate the arank number of Kₙ ☐ Kₘ where n > m.
Keywords: graph colorings, rankings of graphs, minimal rankings, rank number, arank number, Cartesian product of graphs, rook's graph 
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     title = {Minimal rankings of the {Cartesian} product {Kₙ} ☐ {Kₘ}},
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Eyabi, Gilbert; Jacob, Jobby; Laskar, Renu; Narayan, Darren; Pillone, Dan. Minimal rankings of the Cartesian product Kₙ ☐ Kₘ. Discussiones Mathematicae. Graph Theory, Tome 32 (2012) no. 4, pp. 725-735. http://geodesic.mathdoc.fr/item/DMGT_2012_32_4_a8/