发表机构
Instituto de Matemáticas, UNAM; Facultad de Ciencias, UNAM; Escuela Nacional de Estudios Superiores, UNAM; Centro de Ciencias Matemáticas, UNAM（墨西哥国立自治大学数学研究所; 墨西哥国立自治大学理学院; 墨西哥国立自治大学高等研究国立学院; 墨西哥国立自治大学数学科学中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究不连通图的边序可实现性，发现K_n⊔K_n存在边序在ℝ^{n-2}不可实现但在ℝ^{n-1}可实现，还刻画了两环不交并的可实现性，给出对应最大边数的渐近界。
AI 中文摘要
设G为一个图，其边集上赋予全序≺。若存在一种将G的顶点嵌入到ℝ^d中的方式，使得边的欧氏长度恰好诱导出该全序≺，则称≺在ℝ^d中可实现。Almendra-Hernández与Martínez-Sandoval证明，完全图K_n的每条边的全序都可在ℝ^{n-2}中实现。本文表明，对于两个完全图的不交并，情况并非如此：对所有n≥3，存在K_n⊔K_n的边集上的全序，其在ℝ^{n-2}中不可实现，但在ℝ^{n-1}中可实现。令人惊讶的是，不连通图上边序的可实现性并非由其在各连通分支上的限制决定。我们还研究了实直线上的可实现性：刻画了哪些两个环的不交并是可实现的，并估计了n顶点图中，所有边序在实直线上均为可实现时的最大边数。在一般维度下，我们证明，n顶点图中所有边序都可在ℝ^d中实现时，其最大边数为dn+O(dn/ln(dn))。
英文摘要
Let $G$ be a graph together with a total order $\prec$ on its edges. We say that $\prec$ is realizable in $\mathbb{R}^d$ if there is a placement of the vertices of $G$ in $\mathbb{R}^d$ such that the Euclidean lengths of the edges induce exactly the order $\prec$. Almendra-Hernández and Martínez-Sandoval proved that every total order on the edges of the complete graph $K_n$ is realizable in $\mathbb{R}^{n-2}$. We show that the same is not true for the disjoint union of two complete graphs: for every $n\geq 3$ there is a total order on the edges of $K_n\sqcup K_n$ that is not realizable in $\mathbb{R}^{n-2}$, but is in $\mathbb{R}^{n-1}$. Surprisingly, the realizability of an order on a disconnected graph is not determined by its restrictions to the connected components. We also study realizability on the real line: we characterize which disjoint unions of two cycles are realizable, and estimate the largest number of edges an $n$-vertex graph can have while all of its edge-orders remain realizable on the line. In general dimension, we show that the largest number of edges of an $n$-vertex graph all of whose edge-orders are realizable in $\mathbb{R}^d$ is $dn+O\!\left(dn/\ln(dn)\right)$.
