Discussiones Mathematicae Graph Theory 32(4) (2012) 725-735
doi: 10.7151/dmgt.1634

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Minimal Rankings of the Cartesian
Product Kn☐ Km

Gilbert Eyabi(1), Jobby Jacob(2), Renu C. Laskar(3)
Darren A. Narayan(2) and Dan Pillone(4)

(1) Anderson University, Anderson, SC 29621
(2) School of Mathematical Sciences, Rochester Institute of Technology Rochester, NY 14623
(3) Department of Mathematical Sciences, Clemson University Clemson SC 29634
(4) Bally Technologies, USA
e-mail: geyabi@andersonuniversity.edu
jxjsma@rit.edu
rclsk@clemson.edu
dansma@rit.edu
dpillone@hotmail.com

Abstract

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, ψr(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

2010 Mathematics Subject Classification: 05C78, 05C15, 05C76.

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Received 20 June 2011
Revised 20 December 2011
Accepted 29 December 2011