Embedding an Arbitrary Tree in a Graceful Tree

Abstract

A function f is called a graceful labeling of a graph G with m edges if f is an injective function from V(G) to \(\{0,1,2,\ldots ,m\}\) such that when every edge uv is assigned the edge label \(|f(u)-f(v)|\), then the resulting edge labels are distinct. A graph which admits a graceful labeling is called a graceful graph. The popular Graceful Tree Conjecture states that every tree is graceful. The Graceful Tree Conjecture remains open for over four decades. Although there are a few general results and techniques on the construction of graceful trees, settling the conjecture seems to be very hard. In this paper, we have introduced a new and different method of constructing graceful trees from a given arbitrary tree. More precisely, we show that every tree can be embedded in a graceful tree with at most km edges, \(k<\lceil \frac{m}{4} \rceil \), where m is the number of edges of the given arbitrary tree. Further, we pose a related open problem toward settling the Graceful Tree Conjecture.