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The Antimagic Graph Conjecture asserts that every connected graph $G = (V,E)$ except $K_2$ admits an edge labeling such that each label $1, 2, \dots,|E|$ is used exactly once and the sums of the labels on all edges incident to agiven vertex are distinct. On the other extreme, an edge labeling is magic ifthe sums of the labels on all edges incident to each vertex are the same. Inthis paper we approach antimagic labelings by introducing partially magiclabelings, where "magic occurs'' just in a subset of $V$. We generalizeStanley's theorem about the magic graph labeling counting function to theassociated counting function of partially magic labelings and prove that it isa quasi-polynomial of period at most $2$. This allows us to introduce weakantimagic labelings (for which repetition is allowed), and we show that everybipartite graph satisfies a weakened version of the Antimagic Graph Conjecture.

Partially magic labelings and the Antimagic Graph Conjecture