A graceful labeling of a graph with q edges is a labeling of its vertices using the integers in [0, q], such that no two vertices are assigned the same label and each edge is uniquely identified by the absolute difference between the labels of its endpoints. The well known Graceful Tree Conjecture (GTC) states that all trees are graceful, and it remains open. It was proved in 1999 by Broersma a...

A tree of order n is said to be graceful if the vertices can be assigned the labels {0, . . . , n−1} such that the absolute value of the differences in vertex labels between adjacent vertices generate the set {1, . . . , n− 1}. The Graceful Tree Conjecture is the unproven claim that all trees are graceful. We present major results known on graceful trees from those dating from the problem’s ori...

Ryan Jones, Western Michigan University We introduce a modular edge-graceful labeling of a graph a dual concept to the common graceful labeling. A 1991 conjecture known as the Modular Edge-Graceful Tree Conjecture states that every tree of order n where n 6≡ 2 (mod 4) is modular edge-graceful. We show that this conjecture is true. More general results and questions on this topic are presented.

Graceful tree conjecture is a well-known open problem in graph theory. Here we present a computational approach to this conjecture. An algorithm for finding graceful labelling for trees is proposed. With this algorithm, we show that every tree with at most 35 vertices allows a graceful labelling, hence we verify that the graceful tree conjecture is correct for trees with at most 35 vertices.

Graceful labelling is studied on undirected graphs since graceful graphs can be used in some H-decomposition problems. In this note, we investigate the directed graceful problem for many orientations of undirected trees with short diameters, and provide some directed trees that deny any digraceful labelling. AMS Subject Classification (2000): 05C78

Hrnciar and Haviar [3] gave a method to a construct a graceful labeling for all trees of diameter at most five. Based on their method and the methods described in Balbuena et al [1], we shall describe a new construction for gracefully labeled trees by attaching trees to the vertices of a tree admitting a bipartite graceful labeling.

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.