Almost all trees are almost graceful

Abstract

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled 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 > 0 and for all n > n 0 ( ). 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 + )n⌉}. Then there is an injective labeling 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 labeling. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labeling with high probability.

Packing degenerate graphs

“…Generalising in the direction of removing the restriction to bounded degree graphs, Ferber and Samotij [11] showed two near-perfect packing results for trees, one for spanning trees of maximum degree O n 1/6 log −6 n , and one for almost spanning trees of maximum degree O n/ log n . The latter result also follows in the particular case of Ringel's Conjecture from the work of Adamaszek, Allen, Grosu, Hladký [1]. The focus of [1] is the so-called Graceful Tree 1 The Erdős-Sós Conjecture states that if an n-vertex graph has more than 1 2 (k − 1)n edges then it contains each tree of order k + 1.…”

“…The latter result also follows in the particular case of Ringel's Conjecture from the work of Adamaszek, Allen, Grosu, Hladký [1]. The focus of [1] is the so-called Graceful Tree 1 The Erdős-Sós Conjecture states that if an n-vertex graph has more than 1 2 (k − 1)n edges then it contains each tree of order k + 1. A proof (of a slightly weaker form of) the Erdős-Sós Conjecture was announced by Ajtai, Komlós, Simonovits and Szemerédi in the 1990s.…”

Packing trees of unbounded degrees in random graphs

“…. , ϕ s (v j↓−1 ), the images of the d j children of v j form a uniform random d j -element ordered subset of N s j (1). It follows that if N s j (1) = U s j for every j ∈ J, then ϕ s (v 0 ), .…”

“…where the inequality holds as N s j (1) ⊆ U s j . Recall (for example, from the proof of Claim 5.1) that an x ∈ V belongs to N s j (1) if and only if x ∈ U s j and ϕ s (v j ) ∈ N H (x) and hence,…”

“…By Lemma 5.3, conditioned on F s,ji , for each k ∈ I \ {i}, the pair (ϕ s (v i ), ϕ s (v k )) is a uniformly chosen random 2-element ordered subset of N s ji (1). In particular, by (27),…”