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arXiv 1402.6570math.CO
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新的优美直径-6树木的转移

New graceful diameter-6 trees by transfers

Matt Superdock

备注：
This paper is based on my undergraduate senior thesis for the math department at Princeton University, from which I graduated in June 2013
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摘要：

给定图$G$，$G$的标号是内射函数$f:V(G)\rightarrow\mathbb{Z}_{\ge 0}$。在标记$f$下，顶点$v$的标记是$f(v)$，并且边$uv$的诱导标记是$|f(u) - f(v)|$。如果顶点的标号是$\{0, 1, \ldots , |V(G)| - 1\}$，且边的导出标号是不同的，则标号$f$是优美的.图$G$是优美的，如果它有优美标号。优美树猜想，由Kotzig在1960年代后期提出，指出所有的树都是优美的。是否每个直径为6的树都有优美标号是一个公开问题。本文证明了：如果$T$是一棵中心顶点和根都是$v$的树，使得不在最后两层的每个顶点都有奇数个子树，并且$T$满足下列条件（a）-（e）之一，则$T$有优美标号$f$，其中有$f(v) = 0$：（a）$T$是直径为6的完全树;（b）$T$是直径为6的树，使得与$v$的距离为2的两个叶子都不是兄弟，并且与$v$的距离为2的每个叶子都有一个具有偶数个子女的兄弟;（c）$T$是直径- $2r$完全树，使得距离$v$的距离为$r - 1$的顶点的数目，具有偶数个子，不是$3\pmod{4}$ ;（d）$T$是一个直径- $2r$树，使得$r - 1$与$v$的距离（具有偶数个子节点）的顶点数不是$3\pmod{4}$，$r - 1$与$v$的距离没有两个叶子是兄弟姐妹，并且$r - 1$与$v$的距离的每个叶子都有一个具有偶数个子节点的兄弟姐妹;（e）$T$是直径为6的树，使得每个内部顶点具有奇数个子。特别是，所有深度为3的树，其中每个内部顶点有奇数个孩子是优美的。

英文摘要：

Given a graph $G$, a labeling of $G$ is an injective function $f:V(G)\rightarrow\mathbb{Z}_{\ge 0}$. Under the labeling $f$, the label of a vertex $v$ is $f(v)$, and the induced label of an edge $uv$ is $|f(u) - f(v)|$. The labeling $f$ is graceful if the labels of the vertices are $\{0, 1, \ldots , |V(G)| - 1\}$, and the induced labels of the edges are distinct. The graph $G$ is graceful if it has a graceful labeling. The Graceful Tree Conjecture, introduced by Kotzig in the late 1960's, states that all trees are graceful. It is an open problem whether every diameter-6 tree has a graceful labeling. In this paper, we prove that if $T$ is a tree with central vertex and root $v$, such that each vertex not in the last two levels has an odd number of children, and $T$ satisfies one of the following conditions (a)-(e), then $T$ has a graceful labeling $f$ with $f(v) = 0$: (a) $T$ is a diameter-6 complete tree; (b) $T$ is a diameter-6 tree such that no two leaves of distance 2 from $v$ are siblings, and each leaf of distance 2 from $v$ has a sibling with an even number of children; (c) $T$ is a diameter-$2r$ complete tree, such that the number of vertices of distance $r - 1$ from $v$, with an even number of children, is not $3\pmod{4}$; (d) $T$ is a diameter-$2r$ tree, such that the number of vertices of distance $r - 1$ from $v$, with an even number of children, is not $3\pmod{4}$, no two leaves of distance $r - 1$ from $v$ are siblings, and each leaf of distance $r - 1$ from $v$ has a sibling with an even number of children; (e) $T$ is a diameter-6 tree, such that each internal vertex has an odd number of children. In particular, all depth-3 trees of which each internal vertex has an odd number of children are graceful.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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