Let G be an undirected graph with n vertices and m edges. A natural number λ is said to be a magic labeling, positive magic labeling, and fractional positive magic labeling, if the edges can be labeled with nonnegative integers, naturals, and rationals ≥ 1, respectively, so that for each vertex the sum of the labels of incident edges is λ. G is said to be regularizable if it has a positive magic labeling. Denoting the minimum positive magic labeling, the minimum fractional positive magic labeling, and the maximum vertex degree by λ*, λ*, and δ, respectively, we prove that λ* ≤ min {[n/2]δ, 2m, 2λ*). The bound 2m is also derivable from a characterization of regularizable graphs stated by Pulleyblank, and regularizability for graphs with nonbipartite components can be tested via the Bourjolly–Pulleyblank algorithm for testing 2-bicriticality in O(nm) time. We show that using the above bounds and maximum flow algorithms of Ahuja–Orlin–Tarjan regularizability (or 2-bicriticality) can be tested in T(n,m) = O(min{nm + n2 log n2, nm log((n/m) log n2 + 2)}), and λ*, a 2-approximates solution to λ* can be computed in O(T(n,m)log n) time. For dense graphs, T(n,m) can be improved using parallel maximum flow algorithms. We exhibit a family of graphs for which [n/2]δ = 2λ* = 2λ*. Finally, given that the edges in G have nonnegative weights satisfying the triangle inequality, using a capacitated magic labeling solution, we construct a 2-approximate algorithm for the problem of covering all the vertices with optimal set of disjoint even cycles, each covering at least four vertices.
