Products of graceful bipartite graphs

Abstract

A graceful labeling of a graph G with n edges is an assignment to each vertex of G a distinct element of $$\{0,\ldots ,n\}$$ such that if each edge of G inherits the label given by the absolute value of the difference of the labels of its incident vertices, then the set of inherited edge labels is $$\{1,\ldots ,n\}$$. The long-standing graceful labeling conjecture proposes that every tree is graceful. In this paper we present methods to combine graceful bipartite graphs to create new graceful graphs. These methods unify and generalize some well-known results in the graceful labeling literature. Along the way we introduce a new class of graceful trees.