# zbMATH — the first resource for mathematics

Note on group distance magic complete bipartite graphs. (English) Zbl 1284.05122
Summary: A $$\Gamma$$-distance magic labeling of a graph $$G=(V,E)$$ with $$| V|=n$$ is a bijection $$\ell$$ from $$V$$ to an abelian group $$\Gamma$$ of order $$n$$ such that the weight $$w(x)=\sum_{y\in N_G(x)}\ell (y)$$ of every vertex $$x\in V$$ is equal to the same element $$\mu\in\Gamma$$, called the magic constant. A graph $$G$$ is called a group distance magic graph if there exists a $$\Gamma$$-distance magic labeling for every abelian group $$\Gamma$$ of order $$| V(G)|$$.
In this paper we give necessary and sufficient conditions for complete $$k$$-partite graphs of odd order $$p$$ to be $$\mathbb Z_p$$-distance magic. Moreover we show that if $$p\equiv 2$$ $$\pmod 4$$ and $$k$$ is even, then there does not exist a group $$\Gamma$$ of order $$p$$ such that there exists a $$\Gamma$$-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 $$n+m\not\equiv 2$$ $$\pmod 4$$.

##### MSC:
 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C78 Graph labelling (graceful graphs, bandwidth, etc.)
##### Keywords:
graph labeling; abelian group
Full Text:
##### References:
  Arumugam S., Froncek D., Kamatchi N., Distance magic graphs¶a survey, J. Indones. Math. Soc., 2011, Special edition, 11-26 · Zbl 1288.05216  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  Cichacz S., Note on group distance magic graphs G[C 4], Graphs Combin. (in press), DOI: 10.1007/s00373-013-1294-z · Zbl 1294.05135  Combe D., Nelson A.M., Palmer W.D., Magic labellings of graphs over finite abelian groups, Australas. J. Combin., 2004, 29, 259-271 · Zbl 1050.05107  Froncek D., Group distance magic labeling of Cartesian product of cycles, Australas. J. Combin., 2013, 55, 167-174 · Zbl 1278.05210  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 · Zbl 1229.05031
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.