Antimagic Labelings of Weighted and Oriented Graphs∗

A graph G is k–weighted–list–antimagic if for any vertex weighting ω : V (G) → R and any list assignment L : E(G)→ 2R with |L(e)| ≥ |E(G)|+k there exists an edge labeling f such that f(e) ∈ L(e) for all e ∈ E(G), labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on n vertices having no K1 or K2 component is ⌊ 4n 3 ⌋ –weighted–list–antimagic. An oriented graph G is k–oriented–antimagic if there exists an injective edge labeling from E(G) into {1, . . . , |E(G)|+ k} such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on n vertices with no K1 component admits an orientation that is ⌊ 2n 3 ⌋ –oriented–antimagic.