Relating edge-coverings to the classification of -magic graphs

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