A Metaheuristic Approach to the Graceful Labeling Problem

In graph theory, a graceful labeling of a graph G = (V, E) with n vertices and m edges is a labeling of its vertices with distinct integers between 0 and m inclusive, such that each edge is uniquely identified by the absolute difference between its endpoints. In this paper, the well-known graceful labeling problem of graphs is represented as an optimization problem, and an algorithm based on Ant Colony Optimization metaheuristic is proposed for finding its solutions. In this regard, the proposed algorithm is applied to different classes of graphs and the results are compared with the few existing methods inside of different literature. A sample graceful graph is shown in Figure 1. Vertex labels are shown inside the vertex circles, and edge labels are shown in red near the related edges. The name “graceful labeling” is due to Solomon W. Golomb; however, this class of labelings was originally given the name β-labelings by Alex Rosa in a 1967 paper on graph labeling (Rosa, 1967). The computational complexity of the graceful labeling problem is not known, but a related problem called harmonious labeling was shown to be NP-complete (Gallian, 2009). In fact, the graceful labeling problem is rather a well-known example of the problems in NP, which are not known to be NP-complete, and neither known to be in P (Johnson, 2005). Many variations of graph labeling have been introduced in recent years by researchers. Various classes of graphs have been proven mathematically to be graceDOI: 10.4018/jamc.2010100103 International Journal of Applied Metaheuristic Computing, 1(4), 42-56, October-December 2010 43 Copyright © 2010, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. ful or non-graceful. A detailed survey of graph labeling problems and the related results are shown in a survey by Gallian (2009). There is an unproved conjecture that all trees are graceful. Although, it is shown that trees with up to 27 vertices are graceful. It is shown that all cycles Cn are graceful if and only if n ≡ 0 or 3 (mod 4). All wheels Wn, Helms Hn, and Crowns Rn are graceful. The complete graphs Kn are graceful if and only if n ≤ 4. The necessary condition for a windmill Kn (m) (n ≥ 3) to be graceful is that n ≤ 5; a windmill Kn (m) consists of m complete graphs Kn with one common vertex (Gallian, 2009). An example for each class of the graphs mentioned above and their graceful labelings are shown in Figure 2. The graceful labeling problem is to find out whether a given graph is graceful or not, and if it is, how to label the vertices. The process of gracefully labeling a graph is a very tedious and difficult task for many classes of graphs (Eshghi & Azimi, 2004). In the problem literature, many methods are presented for proving gracefulness of different classes of graphs theoretically, but most of them did not use a general method for finding the graceful labeling of the graphs to be studied. These theoretical methods focus on Figure 1. An example of graceful labeling of a graph Figure 2. Examples for some classes of graceful graphs and their graceful labelings: (a) a cycle C7, (b) a tree T10, (c) a wheel W4, (d) a helm H5, (e) a crown R5, (f) a windmill K3 (4) 13 more pages are available in the full version of this document, which may be purchased using the "Add to Cart" button on the product's webpage: www.igi-global.com/article/metaheuristic-approach-gracefullabeling-problem/51677?camid=4v1 This title is available in InfoSci-Journals, InfoSci-Journal Disciplines Computer Science, Security, and Information Technology. Recommend this product to your librarian: www.igi-global.com/e-resources/libraryrecommendation/?id=2