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

• Autorzy: D. Ali Mojdeh
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

graph coloring; defining set; cartesian product

Kategorie tematyczne

• 05C15: Coloring of graphs and hypergraphs
• 05C38: Paths and cycles

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2006
• Tom: 26
• Numer: 1
• Strony: 59-72

Daty

wydano

2006

otrzymano

2004-11-06

poprawiono

2005-09-13

Twórcy

autor

D. Ali Mojdeh

• Department of Mathematics, University of Mazandaran, Babolsar, IRAN, P.O. Box 47416-1467

Bibliografia

• [1] J. Cooper, D. Donovan and J. Seberry, Latin squares and critical sets of minimal size, Austral. J. Combin. 4 (1991) 113-120.
• [2] M. Mahdian and E.S. Mahmoodian, A characterization of uniquely 2-list colorable graph, Ars Combin. 51 (1999) 295-305.
• [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] 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.
• [5] E.S. Mahmoodian, R. Naserasr and M. Zaker, Defining sets in vertex colorings of graphs and Latin rectangles, Discrete Math. (to appear).
• [6] E. Mendelsohn and D.A. Mojdeh, On defining spectrum of regular graph, (submitted).
• [7] D.A. Mojdeh, On conjectures of the defining set of (vertex) graph colourings, Austral. J. Combin. (to appear).
• [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.
• [9] D.B. West, Introduction to Graph Theory (Second Edition) (Prentice Hall, USA, 2001).

Identyfikatory

DOI

10.7151/dmgt.1301