Relating edge-coverings to the classification of [formula omitted]-magic graphs

Abstract

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 Z2k-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 Z2k-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 Z2k-magic for all k≥2 if and only if G has a nowhere-zero 4-flow, and that G is zero-sum Z2k-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 Z2k-magic for all k≥4.