We present several constructions for terraces for cyclic groups that are zigzag terraces, graceful terraces or both. These allow us to complete the classification of imperfect twizzler terraces, unify several previously unrelated constructions of terraces from the literature, find terraces for some non-cyclic abelian groups, and show that for any given r the Oberwolfach Problem with parameters ...

In this paper we study a technique to transform α-labeled trees into ρ-labeled forests. We use this result to prove that the complete graph K2n+1 can be decomposed into these types of forests. In addition we show a robust family of trees that admit ρ-labelings, we use this result to describe the set of all trees for which a ρ-labeling must be found to completely solve Kotzig’s conjecture about ...

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 An...

A p. q graph G = V,E is said to be a square graceful graph ifthere exists an injective function f: V G → 0,1,2,3,... , q such that the induced mapping fp : E G → 1,4,9,... , q 2 defined by fp uv = f u − f v is an injection. The function f is called a square graceful labeling of G. In this paper the square graceful labeling of the caterpillar S X1,X2 ,... ,Xn , the graphs Pn−1 1,2,...n ,mK1,n ∪ ...

A Smarandache-Fibonacci triple is a sequence S(n), n ≥ 0 such that S(n) = S(n − 1) + S(n − 2), where S(n) is the Smarandache function for integers n ≥ 0. Clearly, it is a generalization of Fibonacci sequence and Lucas sequence. Let G be a (p, q)-graph and {S(n)|n ≥ 0} a Smarandache-Fibonacci triple. An bijection f : V (G) → {S(0), S(1), S(2), . . . , S(q)} is said to be a super Smarandache-Fibo...

A graceful labeling of a graphG = (V,E) assigns |V | distinct integers from the set {0, . . . , |E|} to the vertices of G so that the absolute values of their differences on the |E| edges of G constitute the set {1, . . . , |E|}. A graph is graceful if it admits a graceful labeling. The forty-year old Graceful Tree Conjecture, due to Ringel and Kotzig, states that every tree is graceful. We pro...

One of the most famous open problems in graph theory is the Graceful Tree Conjecture, which states that every finite tree has a graceful labeling. In this paper, we define graceful labelings for countably infinite graphs, and state and verify a Graceful Tree Conjecture for countably infinite trees.