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Séminaire Lotharingien de Combinatoire, 78B.86 (2017), 11 pp.

# Matthias Beck and Maryam Farahmand

# Partially Magic Labelings and the Antimagic Graph Conjecture

**Abstract.**
The *Antimagic Graph Conjecture* asserts that every connected
graph *G* = (*V*, *E*) except *K*_{2}
admits an edge labeling such that each
label 1, 2, ..., |*E*| is used exactly once and the sums of the
labels on all edges incident to a given vertex are distinct. On the
other extreme, an edge labeling is *magic* if the sums of the
labels on all edges incident to each vertex are the same. In this
paper we approach antimagic labelings by introducing *partially
magic labelings*, where "magic occurs" just in a subset of *V*. We
generalize Stanley's theorem about the magic graph labeling counting
function to the associated counting function of partially magic
labelings and prove that it is a quasi-polynomial of period at most
2. This allows us to introduce *weak antimagic labelings* (for
which repetition is allowed), and we show that every bipartite graph
satisfies a weakened version of the Antimagic Graph Conjecture.

Received: November 14, 2016.
Accepted: February 17, 2017.
Final version: April 1, 2017.

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