AI 中文总结
该研究证明树中列表距离一致的顶点必在单条路径上，推导得出完全k叉树（除高度2二叉树）的ldc为3，并确定所有蜘蛛图的ldc。
AI 中文摘要
n个顶点的连通图G的标号是一个双射c:V(G)→{1,…,n}；记c(u,v)=|c(u)-c(v)|，若对所有v,w，当d(u,v)<d(u,w)时都有c(u,v)≤c(u,w)，则顶点u是列表距离一致的。所有标号中这类顶点的最大数量为列表距离一致性ldc(G)，由Casselgren和Henricsson提出。我们证明，在树中，任意标号的一致顶点都位于单条路径上，沿该路径的标号构成连续整数块（相对于路径的合适方向递增），路径外顶点不接收该块中的标号。由此推导得出，除高度为2的二叉树外，所有完全k叉树的ldc均为3，且确定了所有蜘蛛图的ldc。
英文摘要
A labeling of a connected graph $G$ on $n$ vertices is a bijection $c:V(G)\to\{1,\dots,n\}$; writing $c(u,v)=|c(u)-c(v)|$, a vertex $u$ is list-distance consistent if $d(u,v)<d(u,w)$ implies $c(u,v)\le c(u,w)$ for all $v,w$. The maximum number of such vertices over all labelings is the list-distance consistency ldc$(G)$, introduced by Casselgren and Henricsson. We prove that in a tree, the consistent vertices of any labeling lie on a single path, along which the labels form a block of consecutive integers in increasing order (with respect to a suitable orientation of the path), no vertex off the path receiving a label from that block. We deduce that ldc equals $3$ for every complete $k$-ary tree except the binary tree of height two, and we determine ldc for all spiders.
