Magic labeling in graphs: Bounds, complexity, and an application to a variant of TSP

Abstract

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 6, 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 + n 2 √log n, nm log((n/m)√logn + 2)}), and λ * , as well as 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.