On the uniqueness of d-vertex magic constant

• Autorzy: S. Arumugam , N. Kamatchi , G.R. Vijayakumar
• Języki publikacji: EN

Abstrakty

EN

Let G = (V,E) be a graph of order n and let D ⊆ {0, 1, 2, 3, . . .}. For v ∈ V, let ND(v) = {u ∈ V : d(u, v) ∈ D}. The graph G is said to be D-vertex magic if there exists a bijection f : V (G) → {1, 2, . . . , n} such that for all v ∈ V, ∑uv∈ND(v) f(u) is a constant, called D-vertex magic constant. O’Neal and Slater have proved the uniqueness of the D-vertex magic constant by showing that it can be determined by the D-neighborhood fractional domination number of the graph. In this paper we give a simple and elegant proof of this result. Using this result, we investigate the existence of distance magic labelings of complete r-partite graphs where r ≥ 4.

Słowa kluczowe

EN

distance magic graph; D-vertex magic graph; magic constant; dominating function; fractional domination numbe

• Wydawca: De Gruyter Open
• Czasopismo: Discussiones Mathematicae Graph Theory
• Rocznik: 2014
• Tom: 34
• Numer: 2
• Strony: 279-286

Daty

wydano

2014-05-01

online

2014-04-12

Twórcy

autor

S. Arumugam

• National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH) Kalasalingam University Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India

autor

N. Kamatchi

• National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH) Kalasalingam University Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India

autor

G.R. Vijayakumar

• Vijayakumar School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road, Colaba, Mumbai 400 005, India

Bibliografia

• [1] S. Arumugam, D. Fronček and N. Kamatchi, Distance magic graphs-A survey, J. Indones. Math. Soc., Special Edition (2011) 11-26.
• [2] S. Beena, On ∑ and ∑′ labelled graphs, Discrete Math. 309 (2009) 1783-1787. doi:10.1016/j.disc.2008.02.038[Crossref]
• [3] G. Chartrand and L. Lesniak, Graphs & Digraphs, 4th Edition (Chapman and Hall, CRC, 2005).
• [4] D. Grinstead and P.J. Slater, Fractional domination and fractional packings in graphs, Congr. Numer. 71 (1990) 153-172.
• [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs, Advanced Topics (Marcel Dekker, Inc., 1998).
• [6] M. Miller, C. Rodger and R. Simanjuntak, Distance magic labelings of graphs, Australas. J. Combin. 28 (2003) 305-315.
• [7] A. O’Neal and P.J. Slater, An introduction to distance D magic graphs, J. Indones. Math. Soc., Special Edition (2011) 91-107.
• [8] A. O’Neal and P.J. Slater, Uniqueness of vertex magic constants, SIAM J. Discrete Math. 27 (2013) 708-716. doi:10.1137/110834421[Crossref][WoS]
• [9] K.A. Sugeng, D. Fronček, M. Miller, J. Ryan and J. Walker, On distance magic labeling of graphs, J. Combin. Math. Combin. Comput. 71 (2009) 39-48.
• [10] V. Vilfred, ∑-labelled graph and circulant graphs, Ph.D. Thesis, University of Kerala, Trivandrum, India, 1994.

Identyfikatory

DOI

10.7151/dmgt.1728