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Graceful labeling is one of the best known labeling methods of graphs. Despite the large number of papers published on the subject of graph labeling, there are few particular techniques to be used by researchers to gracefully label graphs. In this paper, first a new approach based on the mathematical programming technique is presented to model the graceful labeling problem. Then a “branching method” is developed to solve the problem for special classes of graphs. Computational results show the efficiency of the proposed algorithm for different classes of graphs. One of the interesting results of our model is in the class of trees. The largest tree known to be graceful has at most 27 vertices... 

In this paper, an algorithm based on ant colony optimization metaheuristic is proposed for finding solutions to the well-known graceful labeling problem of graphs. Despite the large number of papers published on the theory of this problem, there are few particular techniques introduced by researchers for gracefully labeling graphs. The proposed algorithm is applied to many classes of graphs, and the results obtained have proven satisfactory when compared to those of the existing methods in the literature  

A labeling (or valuation) of a graph G = (V, E) is a map that carries graph elements to numbers (usually to the positive or non-negative integers). The two famous labeling methods are called graceful labeling and edge-magic total labeling. A function f is called a graceful labeling of graph G with m edges if f is an injection from the vertices of G to the set {0, 1,...,m} such that when each edge (x, y) is assigned the label |f(x) - f(y)|, the resulting edge labels are distinct. An edge-magic total labeling is another well-known labeling that was motivated by the notion of magic squares in number theory. Despite the large number of papers published on the subject of graph labeling, there are... 

In this paper, an algorithm based on Ant Colony Optimization metaheuristic is proposed for finding solutions to the well-known graceful labeling problem of graphs. Despite the large number of papers published on the theory of this problem, there are few particular techniques introduced by researchers for gracefully labeling graphs. The proposed algorithm is applied to many classes of graphs, and the results obtained have proven satisfactory when compared to those of the existing methods in the literature. © 2007 IEEE