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

• Volume: 26, Issue: 1, page 59-72
• ISSN: 2083-5892

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## Abstract

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

## How to cite

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D. Ali Mojdeh. "Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph." Discussiones Mathematicae Graph Theory 26.1 (2006): 59-72. <http://eudml.org/doc/270621>.

@article{D2006,
abstract = {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.},
author = {D. Ali Mojdeh},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {graph coloring; defining set; cartesian product},
language = {eng},
number = {1},
pages = {59-72},
title = {Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph},
url = {http://eudml.org/doc/270621},
volume = {26},
year = {2006},
}

TY - JOUR
AU - D. Ali Mojdeh
TI - Defining sets in (proper) vertex colorings of the Cartesian product of a cycle with a complete graph
JO - Discussiones Mathematicae Graph Theory
PY - 2006
VL - 26
IS - 1
SP - 59
EP - 72
AB - 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.
LA - eng
KW - graph coloring; defining set; cartesian product
UR - http://eudml.org/doc/270621
ER -

## References

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1. [1] J. Cooper, D. Donovan and J. Seberry, Latin squares and critical sets of minimal size, Austral. J. Combin. 4 (1991) 113-120. Zbl0759.05017
2. [2] M. Mahdian and E.S. Mahmoodian, A characterization of uniquely 2-list colorable graph, Ars Combin. 51 (1999) 295-305. Zbl0977.05046
3. [3] M. Mahdian, E.S. Mahmoodian, R. Naserasr and F. Harary, On defining sets of vertex colorings of the cartesian product of a cycle with a complete graph, Combinatorics, Graph Theory and Algorithms (1999) 461-467.
4. [4] E.S. Mahmoodian and E. Mendelsohn, On defining numbers of vertex coloring of regular graphs, 16th British Combinatorial Conference (London, 1997). Discrete Math. 197/198 (1999) 543-554. Zbl0924.05025
5. [5] E.S. Mahmoodian, R. Naserasr and M. Zaker, Defining sets in vertex colorings of graphs and Latin rectangles, Discrete Math. (to appear). Zbl0879.05028
6. [6] E. Mendelsohn and D.A. Mojdeh, On defining spectrum of regular graph, (submitted). Zbl1227.05148
7. [7] D.A. Mojdeh, On conjectures of the defining set of (vertex) graph colourings, Austral. J. Combin. (to appear). Zbl1107.05035
8. [8] A.P. Street, Defining sets for block designs; an update, in: C.J. Colbourn, E.S. Mahmoodian (eds), Combinatorics advances, Mathematics and its applications (Kluwer Academic Publishers, Dordrecht, 1995) 307-320. Zbl0837.05017
9. [9] D.B. West, Introduction to Graph Theory (Second Edition) (Prentice Hall, USA, 2001).

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