Almost all trees are almost graceful

TL;DR

This paper proves that almost all trees can be labeled to have pairwise distinct edge differences under relaxed conditions, supporting an approximate version of the Ringel-Kotzig conjecture for these trees.

Contribution

It introduces a relaxation of the Graceful Tree Conjecture and demonstrates that almost all trees admit such labelings, using a probabilistic approach.

Findings

Almost all trees can be labeled with distinct edge differences under certain bounds.

The method allows packing multiple copies of these trees into complete graphs.

Supports an approximate version of the Ringel-Kotzig conjecture for most trees.

Abstract

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labelled by using the numbers {1,2,...,n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each c>0 and for all n>n_0(c). Suppose that (i) the maximum degree of T is bounded by O(n/log n), and (ii) the vertex labels are chosen from the set {1,2,..., (1+c)n}. Then there is an injective labelling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labelling. As a consequence, for any such tree T we can pack (2+2c)n-1 copies of T into the complete graph of order (2+2c)n-1 cyclically. This proves an approximate version of the Ringel-Kotzig conjecture (which asserts the existence of a cyclic packing of 2n-1 copies of any T into the complete graph of order 2n-1) for these trees. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labelling with high probability.