Note on group distance magic complete bipartite graphs

• Autorzy: Sylwia Cichacz
• Języki publikacji: EN

Abstrakty

EN

A Γ-distance magic labeling of a graph G = (V, E) with |V| = n is a bijection ℓ from V to an Abelian group Γ of order n such that the weight $$w(x) = \sum\nolimits_{y \in N_G (x)} {\ell (y)}$$ of every vertex x ∈ V is equal to the same element µ ∈ Γ, called the magic constant. A graph G is called a group distance magic graph if there exists a Γ-distance magic labeling for every Abelian group Γ of order |V(G)|. In this paper we give necessary and sufficient conditions for complete k-partite graphs of odd order p to be ℤp-distance magic. Moreover we show that if p ≡ 2 (mod 4) and k is even, then there does not exist a group Γ of order p such that there exists a Γ-distance labeling for a k-partite complete graph of order p. We also prove that K m,n is a group distance magic graph if and only if n + m ≢ 2 (mod 4).

Słowa kluczowe

EN

Graph labeling; Abelian group

Kategorie tematyczne

• 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
• 05C78: Graph labelling (graceful graphs, bandwidth, etc.)

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2014
• Tom: 12
• Numer: 3
• Strony: 529-533

Daty

wydano

2014-03-01

online

2013-12-21

Twórcy

autor

Sylwia Cichacz

• AGH University of Science and Technology

Bibliografia

• [1] Arumugam S., Froncek D., Kamatchi N., Distance magic graphs¶a survey, J. Indones. Math. Soc., 2011, Special edition, 11–26
• [2] Beena S., On Σ and Σ′ labelled graphs, Discrete Math., 2009, 309(6), 1783–1787 http://dx.doi.org/10.1016/j.disc.2008.02.038
• [3] Cichacz S., Note on group distance magic graphs G[C 4], Graphs Combin. (in press), DOI: 10.1007/s00373-013-1294-z
• [4] Combe D., Nelson A.M., Palmer W.D., Magic labellings of graphs over finite abelian groups, Australas. J. Combin., 2004, 29, 259–271
• [5] Froncek D., Group distance magic labeling of Cartesian product of cycles, Australas. J. Combin., 2013, 55, 167–174
• [6] Kaplan G., Lev A., Roditty Y., On zero-sum partitions and anti-magic trees, Discrete Math., 2009, 309(8), 2010–2014 http://dx.doi.org/10.1016/j.disc.2008.04.012

Identyfikatory

DOI

10.2478/s11533-013-0356-z