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Minimal rankings of the Cartesian product Kₙ ☐ Kₘ

Gilbert EyabiJobby JacobRenu C. LaskarDarren A. NarayanDan Pillone — 2012

Discussiones Mathematicae Graph Theory

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.

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