# Minimal rankings of the Cartesian product Kₙ ☐ Kₘ

• Volume: 32, Issue: 4, page 725-735
• ISSN: 2083-5892

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## Abstract

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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, ${\psi }_{r}\left(G\right)$, 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.

## How to cite

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Gilbert Eyabi, et al. "Minimal rankings of the Cartesian product Kₙ ☐ Kₘ." Discussiones Mathematicae Graph Theory 32.4 (2012): 725-735. <http://eudml.org/doc/270880>.

@article{GilbertEyabi2012,
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, $ψ_r(G)$, 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.},
author = {Gilbert Eyabi, Jobby Jacob, Renu C. Laskar, Darren A. Narayan, Dan Pillone},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph colorings; rankings of graphs; minimal rankings; rank number; arank number; Cartesian product of graphs; rook's graph},
language = {eng},
number = {4},
pages = {725-735},
title = {Minimal rankings of the Cartesian product Kₙ ☐ Kₘ},
url = {http://eudml.org/doc/270880},
volume = {32},
year = {2012},
}

TY - JOUR
AU - Gilbert Eyabi
AU - Jobby Jacob
AU - Darren A. Narayan
AU - Dan Pillone
TI - Minimal rankings of the Cartesian product Kₙ ☐ Kₘ
JO - Discussiones Mathematicae Graph Theory
PY - 2012
VL - 32
IS - 4
SP - 725
EP - 735
AB - 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.
LA - eng
KW - graph colorings; rankings of graphs; minimal rankings; rank number; arank number; Cartesian product of graphs; rook's graph
UR - http://eudml.org/doc/270880
ER -

## References

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