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...

We exhibit a graceful labelling for each generalised Petersen graph P8t,3 with t ≥ 1. As an easy consequence, we obtain that for any fixed t the corresponding graph is the unique starter graph for a cyclic edgedecomposition of the complete graph K2t+1. Due to its extreme versatility, the technique employed looks promising for finding new graceful labellings, not necessarily involving generalise...

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.

Symmetry in a Constraint Satisfaction Problem can cause wasted search, which can be avoided by adding constraints to the CSP to exclude symmetric assignments or by modifying the search algorithm so that search never visits assignments symmetric to those already considered. One such approach is SBDS (Symmetry Breaking During Search); a modification is GAP-SBDS, which works with the symmetry grou...

The Graceful Tree Conjecture is getting old – though 40 years are not so many – while researchers form all over the world keep on trying to put an affirmative end to it. Kotzig called a disease the effort of proving it. In this paper we fall into the opposite disease, namely by moving towards the search of a tree that is not graceful. Our first result, on suitable attachments of graceful trees,...

Symmetry in a Constraint Satisfaction Problem (CSP) can cause wasted search, which can be avoided by adding constraints to the CSP to exclude symmetric assignments or by modifying the search algorithm so that search never visits assignments symmetric to those already considered. Two such approaches are SBDS (Symmetry Breaking During Search) and SBDD (Symmetry Breaking by Dominance Detection); m...