### 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 λ̂(Black star), λ̂_{*}, and δ, respectively, we prove that λ̂(Black star) ≤ min{[n/2]δ, 2m, 2λ̂_{*}}. 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)√log n + 2)}), and λ̂_{*}, as well as a 2-approximates solution to λ̂(Black star) 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 = 2λ̂(Black star) = 2λ̂_{*}. 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.

Original language | English (US) |
---|---|

Pages (from-to) | 211-219 |

Number of pages | 9 |

Journal | Networks |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1996 |

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### All Science Journal Classification (ASJC) codes

- Software
- Information Systems
- Hardware and Architecture
- Computer Networks and Communications

### Cite this

*Networks*,

*28*(4), 211-219. https://doi.org/10.1002/(SICI)1097-0037(199612)28:4<211::AID-NET5>3.0.CO;2-O

}

*Networks*, vol. 28, no. 4, pp. 211-219. https://doi.org/10.1002/(SICI)1097-0037(199612)28:4<211::AID-NET5>3.0.CO;2-O

**Magic labeling in graphs : Bounds, complexity, and an application to a variant of TSP.** / Kalantari, Bahman; Khosrovshahi, G. B.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Magic labeling in graphs

T2 - Bounds, complexity, and an application to a variant of TSP

AU - Kalantari, Bahman

AU - Khosrovshahi, G. B.

PY - 1996/1/1

Y1 - 1996/1/1

N2 - 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 λ̂(Black star), λ̂*, and δ, respectively, we prove that λ̂(Black star) ≤ min{[n/2]δ, 2m, 2λ̂*}. 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 n, nm log((n/m)√log n + 2)}), and λ̂*, as well as a 2-approximates solution to λ̂(Black star) 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 = 2λ̂(Black star) = 2λ̂*. 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.

AB - 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 λ̂(Black star), λ̂*, and δ, respectively, we prove that λ̂(Black star) ≤ min{[n/2]δ, 2m, 2λ̂*}. 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 n, nm log((n/m)√log n + 2)}), and λ̂*, as well as a 2-approximates solution to λ̂(Black star) 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 = 2λ̂(Black star) = 2λ̂*. 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=0038879425&partnerID=8YFLogxK

U2 - 10.1002/(SICI)1097-0037(199612)28:4<211::AID-NET5>3.0.CO;2-O

DO - 10.1002/(SICI)1097-0037(199612)28:4<211::AID-NET5>3.0.CO;2-O

M3 - Article

VL - 28

SP - 211

EP - 219

JO - Networks

JF - Networks

SN - 0028-3045

IS - 4

ER -