# DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

# IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

# Discussiones Mathematicae Graph Theory

## 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