完全二叉树和蜘蛛树优美标号中的交替极值
Alternating Extremes in Graceful Labelings of Full Binary Trees and Spider Trees
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中文总结 AI 辅助
研究完全二叉树和蜘蛛树优美标号中固定形式的交替极值问题，证明了梳状完全二叉树的固定脊柱猜想并计算验证相关完全二叉树，证明蜘蛛树自匹配腿填充定理，给出相关计算结果并表述六臂问题。
中文摘要 AI 辅助
我们研究了一种固定形式的优美标号。对于完全二叉树，我们探讨是否存在一些从根到叶的最深路径能呈现交替极值模式\(0,n - 1,1,n - 2,\dots\)。这样的脊柱使用了极值标签和最大差值，迫使所有非脊柱顶点和边分别使用中间标签和较小差值。我们证明了梳状完全二叉树的固定脊柱猜想，通过计算验证了所有阶数为\(23\)的非同构有根完全二叉树，还给出了一个例子表明固定脊柱标号不一定能选为\(\alpha\)-标号。对于蜘蛛树，我们证明了自匹配腿的填充定理：基于中心标签\(1\)的两两不相交的腿，其中至少有一条包含标签\(0\)，可以组合成一个优美的蜘蛛树，未使用的标签作为中心叶附着。这为具有足够多叶的混合长度蜘蛛树产生了优美标号。我们还报告了使用按最大未使用差值排序的深度优先搜索的计算结果，并将六臂问题表述为偏移五臂剩余问题。
英文摘要
We study a pinned form of graceful labeling. For full binary trees, we ask whether some deepest root-to-leaf path can carry the alternating extreme pattern $0,n-1,1,n-2,\dots$. Such a spine uses the extreme labels and largest differences, forcing all off-spine vertices and edges to use the middle labels and smaller differences, respectively. We prove this pinned-spine conjecture for comb full binary trees, verify it computationally for all rooted non-isomorphic full binary trees through order $23$, and give an example showing that a pinned-spine labeling cannot always be chosen as an $α$-labeling. For spider trees, we prove a packing theorem for self-matched legs: pairwise disjoint legs based at hub label $1$, at least one of which contains label $0$, can be combined into a graceful spider, with unused labels attached as hub leaves. This yields graceful labelings for mixed-length spiders with sufficiently many leaves. We also report computations using a depth-first search ordered by largest unused differences and formulate the six-arm problem as an offset five-arm residual problem.
