Constructing Graceful Graphs with Caterpillars

Document Type: Research Paper

Authors

Department of Mathematics, Clayton State University, Morrow, Georgia 30260, USA

Abstract

A graceful labeling of a graph G of size n is an injective assignment of integers from {0, 1,..., n} to the vertices of G, such that when each edge of G has assigned a weight, given by the absolute di erence of the labels of its end vertices, the set of weights is {1, 2,..., n}. If a graceful labeling f of a bipartite graph G assigns the smaller labels to one of the two stable sets of G, then f is called an -labeling and G is said to be an α-graph. A tree is a caterpillar if the deletion of all its leaves results in a path. In this work we study graceful labelings of the disjoint union of a cycle and a caterpillar. We present necessary conditions for this union to be graceful and, in the case where the cycle has even size, to be an α -graph. In addition, we present a new family of graceful trees constructed using α-labeled caterpillars.

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