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Home > Journals > TAG > Vol. 4 > Iss. 1 (2017)

 

Publication Date

2017

Abstract

A graceful labeling of a graph G of size n is an injective assignment of integers from the set {0,1,…,n} to the vertices of G such that when each edge has assigned a weight, given by the absolute value of the difference of the labels of its end vertices, all the weights are distinct. A graceful labeling is called an α-labeling when the graph G is bipartite, with stable sets A and B, and the labels assigned to the vertices in A are smaller than the labels assigned to the vertices in B. In this work we study graceful and α-labelings of graphs. We prove that the Cartesian product of two α-trees results in an α-tree when both trees admit α-labelings and their stable sets are balanced. In addition, we present a tree that has the property that when any number of pendant vertices are attached to the vertices of any subset of its smaller stable set, the resulting graph is an α-tree. We also prove the existence of an α-labeling of three types of graphs obtained by connecting, sequentially, any number of paths of equal size.

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Creative Commons Attribution 4.0 License
This work is licensed under a Creative Commons Attribution 4.0 License.

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