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Graceful Labeling Algorithms and Complexity - a Survey J. Indones. Math. Soc. Special Edition (2011), pp. 1–9. GRACEFUL LABELING ALGORITHMS AND COMPLEXITY - A SURVEY S. Arumugam1,2, Jay Bagga3 1National Centre for Advanced Research in Discrete Mathematics (n-CARDMATH), Kalasalingam University, Anand Nagar, Krishnankoil-626190, India, [email protected] 2School of Electrical Engineering and Computer Science, The University of Newcastle, NSW 2308, Australia. 3Department of Computer Science, Ball State University, Muncie, IN 47306, USA, [email protected] Abstract. Graceful graphs were first studied by Rosa in 1966. The Kotzig-Ringel graceful tree conjecture states that every tree has a graceful labeling. Aldred and McKay and others [6, 15, 20] have used computer programs to show that trees of order up to 35 are graceful. Bagga et al. investigated algorithms for generating all graceful labelings of certain known classes of graceful graphs, including paths, cycles, and certain other classes of unicyclic graphs. The data generated by such algorithms has led to the discovery of new properties of such graceful labelings. In this paper we present a survey of graceful graph labeling algorithms and related complexity issues. Key words: Graph labelings, graceful labeling, algorithms, complexity. Abstrak. Graf-graf graceful pertama kali dipelajari oleh Rosa pada tahun 1966. Konjektur pohon graceful Kotzig-Ringel menyatakan bahwa setiap pohon memi- liki suatu pelabelan graceful. Aldred dan McKay (1998), Fang (2011), dan Horton (2003) telah menggunakan program komputer untuk menunjukkan bahwa pohon dengan orde hingga 35 adalah graceful. Bagga dkk. (2007) telah menyelidiki algo- ritma untuk menghasilkan semua pelabelan graceful dari graf-graf graceful untuk kelas-kelas graf tertentu seperti lintasan, lingkaran, dan kelas tertentu lainnya dari graf unicyclic. Data yang dihasilkan oleh algoritma tersebut menuntun pada pene- muan sifat baru dari pelabelan graceful. Pada artikel ini kami menyajikan sebuah survei dari algoritma pelabelan graf graceful dan isu-isu kompleksitas terkait. Kata kunci: Pelabelan graf, pelabelan graceful, algoritma, kompleksitas. 2000 Mathematics Subject Classification: 05C30, 05C78, 05C85. Received: 09-08-2011, revised: 09-09-2011, accepted: 04-12-2011. 1 2 S. Arumugam and J. Bagga 1. Introduction For a graph G, we are interested in labeling the vertices and/or edges of G with nonnegative integers. More formally, a labeling of a graph G =(V,E) is a one- one mapping f : V → X, where X is a (finite) set of nonnegative integers. Such a vertex labeling f induces an edge labeling, where an edge e = uv gets the label |f(u) − f(v)|. Suppose that G has q edges. We are interested in labelings where the set of possible labels X is {0, 1, 2,...,q} or {0, 1, 2,..., 2q}. Rosa [23] called such a labeling a valuation. He considered four special types of valuations. Let f be a valuation of a graph G with q edges. (1) f is called an α-valuation if X ⊆ {0, 1, 2,...,q}, the induced edge labels are 1, 2,...,q and there exists an x ∈ {0, 1, 2,...,q} such that for any edge uv of G, either f(u) ≤ x<f(v) or f(v) ≤ x<f(u). (2) f is called a β-valuation if X ⊆ {0, 1, 2,...,q} and the induced edge labels are 1, 2,...,q. (3) f is called a σ-valuation if X ⊆ {0, 1, 2,..., 2q} and the induced edge labels are 1, 2,...,q. (4) f is called a ρ-valuation if X ⊆ {0, 1, 2,..., 2q} and the induced edge labels are x1,x2,...,xq where, xi = i or xi = 2q + 1 − i for each 1 ≤ i ≤ q. Observe that the above definitions are from the strongest to weakest in the sense that every α-valuation is also a β-valuation, every β-valuation is also a σ- valuation, and every σ-valuation is also a ρ-valuation. Golomb [18] introduced the term graceful labeling for a β-valuation. A graceful graph is one that has a graceful labeling. Rosa’s paper [23] generated much research activity in this area and led to a large number of publications. A substantial part of this research effort has been devoted to finding new families of graceful graphs. Several variations of the concept have also been studied. We refer the reader to Gallian’s ongoing survey [16] for an excellent account of research in this area. One of the foremost open problems is the Ringel-Kotzig conjecture that every tree is graceful. As noted by Gallian [16], Aldred and McKay proved, with the aid of a computer program, that all trees with at most 27 vertices are graceful. This has been extended to trees of order up to 35 [15, 20]. It is also known that paths, caterpillars and several other subclasses of trees are graceful. In this paper we present a survey of graceful graph labeling algorithms and related complexity issues. In Section 2, we discuss computational efforts aimed at verifying the graceful tree conjecture. In Section 3, we look at complexity issues in graceful graphs. In Section 4, we survey recent results on generating all graceful labelings of certain families of unicyclic graphs. In Section 4, we discuss other computational and mathematical methods for determining gracefulness. Finally, we conclude in Section 5 with a summary and some open problems. Graceful Labeling Algorithms 3 2. Graceful Tree Conjecture Rosa considered the above valuations in order to investigate decompositions of graphs into trees. For the sake of completeness, we include a brief discussion here. A decomposition of a graph G =(V,E) can be considered to be a partition of the edge set E into subsets E1,E2,...,Er. If Hi is the subgraph of G induced by Ei, then we also say the G decomposes into subgraphs {Hi}. If the subgraphs Hi are all isomorphic to a single graph (say) H then we say that G is H − decomposable and we write H|G. In 1963, Ringel [22] made the following conjecture. Conjecture 2.1. If T is a tree with m edges, then K2m+1 decomposes into 2m + 1 copies of T . Rosa [23] proved the following theorem. Theorem 2.2. If a tree T with m edges has a graceful labeling, then K2m+1 de- composes into 2m + 1 copies of T . In fact, Rosa proved more. A decomposition of Kn by r copies of a subgraph H is called a cyclic decomposition if, when Kn is drawn appropriately with its vertices on a regular polygon, the r copies of H can be obtained by rotations of an appropriate copy of H. Rosa [23] proved the following results. Theorem 2.3. Let G be a graph with m edges. Then G has a ρ- valuation if and only if K2m+1 has a cyclic decomposition into 2m + 1 copies of G. Theorem 2.4. Let G be a graph with m edges, and k ≥ 1 be an integer. If G has an α- valuation then K2km+1 has a cyclic decomposition into k(2km + 1) copies of G. Conjecture 2.1 was followed by the following stronger conjecture, which is known as the Kotzig-Ringel graceful tree conjecture. Conjecture 2.5. Every tree has a graceful labeling. A substantial amount of research effort has been spent towards finding a proof of this conjecture. A number of special classes of trees have been shown to be graceful. These include paths, caterpillars, lobsters with perfect matchings [21, 23] and several others. Rosa [23] showed that all trees with less than 16 edges are graceful, and all trees of up to 4 leaves are graceful. Theorem 2.6. [6, 15, 20] Every tree of order up to 35 is graceful. Aldred and McKay used a computer search to prove the above result. They described this search algorithm as a combination of hill climbing and tabu search. In the same paper they also showed that all trees of at most 27 vertices are har- monious ([19]). Horton [20] used a random backtracking algorithm to show that trees of order up to 29 are graceful. To enumerate trees with a given number of vertices, the algorithm proposed by Wright et al [26] is used. Fang [15] describes a computational approach and shows that every tree of order up to 35 is graceful. 4 S. Arumugam and J. Bagga Verification of gracefulness for trees with 33, 34 or 35 vertices is accomplished with the help of a volunteer computing community, where the computational task is divided into small fragments that are carried out on heterogeneous machines. It is estimated that to verify the gracefulness on a single machine, it may take up to 7.7 years on a Core 2 Duo T7200 computer. The following result of Erd¨os[18] is well-known. See [19] for a proof. Theorem 2.7. Almost all graphs are not graceful. However, the research on the Kotzig-Ringel graceful tree conjecture and on finding other families of graceful graphs continues. 3. Generating and Enumerating Graceful Labelings We now discuss bounds on the numbers of graceful labelings for some classes of graphs. For paths, let G(n) denote the number of graceful labelings of Pn. Aldred et al [7] proved the following result. Theorem 3.1. G(n) = Ω((5/3)n). Adamaszek [5] studied graceful labelings in the context of permutations. He defined a graceful n-permutation as a permutation [σ(0),...,σ(n − 1)] of the set {0, 1,...,n − 1} such that {|σ(1) − σ(0)|, |σ(2) − σ(1)|,..., |σ(n − 1) − σ(n − 2)|} = {1,...,n − 1}. For example, [0, 6, 1, 5, 2, 4, 3] is a graceful n-permutation. Thus a graceful n-permutation of the set {0, 1,...,n − 1} can be identified with a graceful labeling of Pn.
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