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Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph

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Mojdeh D.

Discussiones Mathematicae Graph Theory

2006

tom 26

nr 1

59-72

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 $d(K₃×C_{2r},4)$, which is given an upper and lower bounds for this defining number. Also some bounds of defining number are introduced.

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1 Discussiones Mathematicae Graph Theory

1 Mojdeh D.

1 2006

Wyniki wyszukiwania

Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph