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2014 | 34 | 2 | 279-286
Tytuł artykułu

On the uniqueness of d-vertex magic constant

Treść / Zawartość
Warianty tytułu
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.
Wydawca
Rocznik
Tom
34
Numer
2
Strony
279-286
Opis fizyczny
Daty
wydano
2014-05-01
online
2014-04-12
Twórcy
autor
  • National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH) Kalasalingam University Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India, s.arumugam.klu@gmail.com
autor
  • National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH) Kalasalingam University Anand Nagar, Krishnankoil-626 126, Tamil Nadu, India, n_kamatchi@yahoo.com
  • Vijayakumar School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road, Colaba, Mumbai 400 005, India, vijay@math.tifr.res.in
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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_7151_dmgt_1728
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