# Minimal rankings of the Cartesian product Kₙ ☐ Kₘ

Gilbert Eyabi; Jobby Jacob; Renu C. Laskar; Darren A. Narayan; Dan Pillone

Discussiones Mathematicae Graph Theory (2012)

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

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topGilbert 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 - Renu C. Laskar

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 -

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