Relating edge-coverings to the classification of Z(2)(k)-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 phi from E into A - {0} such that for some c is an element of A, Sigma(e is an element of E(v))phi(e) = c for every v is an element of V, where E(v) is the set of edges incident to v. Additionally, G is zero-sum A-magic if and only if phi 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. (C) 2012 Elsevier B.V. All rights reserved.