In a given graph G = (V,E), a set of vertices S with an assignment of colors to them is said to be a defining set of the vertex coloring of G, if there exists a unique extension of the colors of S to a c ≥ χ(G) coloring of the vertices of G. A defining set with minimum cardinality is called a minimum defining set and its cardinality is the defining number, denoted by d(G,c). The d(G = Cₘ × Kₙ, χ(G)) has been studied. In this note we show that the exact value of defining number d(G = Cₘ × Kₙ, c) with c > χ(G), where n ≥ 2 and m ≥ 3, unless the defining number , which is given an upper and lower bounds for this defining number. Also some bounds of defining number are introduced.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1301,
author = {D. Ali Mojdeh},
title = {Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph},
journal = {Discussiones Mathematicae Graph Theory},
volume = {26},
year = {2006},
pages = {59-72},
zbl = {1102.05023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1301}
}
D. Ali Mojdeh. Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph. Discussiones Mathematicae Graph Theory, Tome 26 (2006) pp. 59-72. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1301/
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