Towards Optimal Embedding of an Arbitrary Tree in a Graceful Tree

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,⋯,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 and notorious Graceful Tree Conjecture: “All trees are graceful”, remains unsettled over four decades inspite of many rigorous attempts and investigations. Inspired by the interesting result of Acharya et al. [B.D. Acharya, S.B. Rao, S. Arumugam, Embedding and NP-Complete problems for Graceful Graphs, Labelings of Discrete Structures and Applications, in: B.D. Acharya, S. Arumugam, Alexander Rosa (Eds.), Narosa Publishing House, New Delhi, 2008, pp. 57–62], in this paper we show that every tree can be embedded in a graceful tree. Practically, we observe that any tree with m edges can be embedded in a graceful tree with less than 2m edges and we discuss a related open problem towards settling the Graceful Tree Conjecture through the embedding approach.