The University of Montana

Department of Mathematical Sciences

Technical report #16/2007

Magic squares and antimagic graphs

**P. Mark Kayll**

Department of Mathematical Sciences

University of Montana

Missoula MT 59812-0864, USA

mark.kayll@umontana.edu

**Jennifer McNulty**

Department of Mathematical Sciences

University of Montana

Missoula MT 59812-0864, USA

jenny.mcnulty@umontana.edu

**James Mihalisin
**Department of Mathematical Sciences

University of Montana

Missoula MT 59812-0864, USA

mihalisi@mso.umt.edu

**Abstract**

An *antimagic labeling* of a graph with *m* edges is a bijection from its edge set to such that the vertex sums are distinct, a vertex sum being the sum of
the -values on the edges incident with the vertex. An *antimagic* graph is one that
admits such a labeling. It was conjectured in [N. Hartsfield and G. Ringel, Supermagic
and antimagic graphs, *J. Recreational Math.* **21** (1989), 107–115] that every connected
graph other than K_{2} is antimagic. We verify this conjecture constructively for a class
of graphs derived from the complete graphs K_{n} = (*V*,*E*) using a variant of magic
squares. In support of our main result, we establish that K* _{n}*, with , possesses a
property stronger than being antimagic: for every function , there exists a
bijectionsuch that the sums , for , are
all different. This ‘robust’ property of

**Keywords:** antimagic, latin square, magic square, RC-magic, transversal system

**AMS Subject Classification:** Primary 05C78 Secondary 05C50, 05B15

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