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The graceful labeling problem is a famous open problem in mathematics and computer science, first described by Alexander Rosa in 1967. The object of the problem is given a graph, is there a way to label the vertices of that graph uniquely with the numbers 0 to m, where m is the size of the graph, such that when its edges are labeled with the absolute differences of the vertex labels, the edges are labeled uniquely? Many different classes of graphs are conjectured to be graceful, the most famous being trees. This paper explores another class of graphs, connected cubic graphs, conjectured to be graceful by El-Zanati and Wannasit (2011), and shows that this is true for cases of graphs with 16 vertices or fewer. To do this, we use a backtracking algorithm to explore cubic graphs and other interesting classes of regular graphs.
Clare Kennedy, George, "An Algorithmic Exploration of Gracefully Labeling Cubic Graphs" (2021). Mathematics, Statistics, and Computer Science Honors Projects. 54.
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