关于树的优美度
On the gracesize of trees
AI总结：
该研究证明了渐近版本的优美树猜想，即足够大的树存在几乎优美的标号，其优美度至少为顶点数的(1-ε)倍。
摘要：
若n个顶点的树T存在双射标号φ:V(T)→{1,…,n}，使得边差{|φ(x)-φ(y)|:xy∈E(T)}两两不同，则称T是优美的。1960年代Rósa提出的长期悬而未决的优美树猜想断言：所有树都是优美的。n个顶点树T的优美度记为gs(T)，是所有双射标号φ下不同边差的最大可能数量，因此优美树猜想等价于：对所有n个顶点的树，gs(T)=n-1。我们证明了该猜想的渐近版本：对任意ε>0，存在n₀，使得每个顶点数n>n₀的树T满足gs(T)≥(1-ε)n，即每个足够大的树都存在几乎优美的标号。
英文摘要：
An $n$-vertex tree $T$ is said to be $\textit{graceful}$ if there exists a bijective labelling $ϕ:V(T)\to \{1,\ldots,n\}$ such that the edge-differences $\{|ϕ(x)-ϕ(y)| : xy\in E(T)\}$ are pairwise distinct. The longstanding graceful tree conjecture, posed by Rósa in the 1960s, asserts that every tree is graceful. The $\textit{graceful}$ of an $n$-vertex tree $T$, denoted $\operatorname{gs}(T)$, is the maximum possible number of distinct edge-differences over all bijective labellings $ϕ:V(T)\to \{1,\ldots,n\}$. The graceful tree conjecture is therefore equivalent to the statement that $\operatorname{gs}(T)=n-1$ for all $n$-vertex trees. We prove an asymptotic version of this conjecture by showing that for every $\varepsilon>0$, there exists $n_0$ such that every tree on $n>n_0$ vertices satisfies $\operatorname{gs}(T)\geqslant (1-\varepsilon)n$. In other words, every sufficiently large tree admits an almost graceful labelling.
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