Relating edge-coverings to the classification of Z2k-magic graphs

Let G=(V,E)G=(V,E) be a finite graph and let (A,+)(A,+) be an abelian group with identity 0. Then GG is AA-magic   if and only if there exists a function ϕϕ from EE into A−{0}A−{0} such that for some c∈Ac∈A, ∑e∈E(v)ϕ(e)=c∑e∈E(v)ϕ(e)=c for every v∈Vv∈V, where E(v)E(v) is the set of edges incident to vv. Additionally, GG is zero-sum A-magic   if and only if ϕϕ exists such that c=0c=0. In this paper, we explore Z2k-magic graphs in terms of even edge-coverings, graph parity, factorability, and nowhere-zero 4-flows. We prove that the minimum kk such that bridgeless GG is zero-sum Z2k-magic is equal to the minimum number of even subgraphs that cover the edges of GG, known to be at most 3. We also show that bridgeless GG is zero-sum Z2k-magic for all k≥2k≥2 if and only if GG has a nowhere-zero 4-flow, and that GG is zero-sum Z2k-magic for all k≥2k≥2 if GG is Hamiltonian, bridgeless planar, or isomorphic to a bridgeless complete multipartite graph. Finally, we establish equivalent conditions for graphs of even order with bridges to be Z2k-magic for all k≥4k≥4.