Efficient Enumeration of Graceful Permutations

A graceful n-permutation is a graceful labeling of an n-vertex path Pn. In this paper we improve the asymptotic lower bound on the number of such permutations from Ω((5/3)n) to Ω(2.37n). This is a computer-assisted proof based on an effective algorithm that enumerates graceful n-permutations. Our algorithm is also presented in detail. 1. Graceful graphs and permutations Let G = (V,E) be an undirected graph with |V | = n and |E| = m. We say that a vertex labeling f : V 7→ N together with an edge labeling g : E 7→ N are a graceful labeling of G if: • f(V ) ⊂ {0, . . . ,m} and f is one-to-one (injective) • g(E) = {1, . . . ,m} • g(uv) = |f(u)− f(v)| for every two vertices u, v ∈ V such, that uv ∈ E Graceful labelings of graphs have received a lot of attention; see [3] for an extensive survey. In this paper we concentrate on the single case when G = Pn is the n-vertex path. Note, that in this case m = n − 1, thus the vertex labels are in bijection with the set {0, . . . , n− 1}. This justifies the following definition: Definition 1. A permutation [σ(0), . . . , σ(n − 1)] of the set {0, 1, . . . , n − 1} is a graceful n-permutation if {|σ(1) − σ(0)|, |σ(2) − σ(1)|, . . . , |σ(n − 1)− σ(n− 2)|} = {1, . . . , n− 1} For instance, [0, 6, 1, 5, 2, 4, 3] is a graceful 7-permutation. The values of a graceful n-permutation can be identified with the vertex labels in some graceful labeling of Pn and vice versa. We shall use these notions interchangeably. Denote by G(n) the number of graceful n-permutations. The sequence G(n) is not well known, not even asymptotically. It has number A006967 in the Sloane’s Online Encyclopedia of Integer Sequences ([4]) where the first 20 terms are listed. Its growth is exponential as shown in [2] and [1]. In the latter the best known estimate, G(n) = Ω(( 3 )n) is proved. Here we shall improve this result by proving the following: Theorem 1. G(n) = Ω(2.37n) This paper is organized as follows. In the next section we introduce a recursive algorithm for the computation of G(n). Next we observe how its efficiency can be vastly improved using some knowledge of the structure of graceful permutations. In 1991 Mathematics Subject Classification. Primary 05C78, Secondary 11Y55. ∗ Warsaw University, email: [email protected].