发表机构
Department of Informatics & Telecommunications, National & Kapodistrian University of Athens; Athens University of Economics and Business（雅典国立卡波迪斯特里阿斯大学信息与通信系; 雅典经济与商业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文改进了$4$-正则图欧拉定向数量的上界，对双连通多重图给出紧界$2^n+2$，对简单图给出指数级上界，并提出分治算法精确计算可分离图，无需穷举。
AI 中文摘要
$4$-正则无向图（简单图或多重图）$G=(V,E)$，其中$n=|V|$为顶点数，其欧拉定向是指对每条边指定方向，使得每个顶点$v \in V$的入度和出度相等。本文改进了某些连通、无环$4$-正则图类的欧拉定向数量的界。此前的界由M. Las Vergnas（1983）给出，精确为$9\cdot 2^{n-3}$，这是对$n\geq 4$的某类多重图的紧界。在此，我们证明所有双连通$4$-正则多重图的欧拉定向数量至多为$2^n+2$，该界也是紧的。我们构造了达到此最大值的图族。对于简单图，我们证明了双连通情形下的上界为$\mathcal{O}(3^{n/2})$，可分离情形下的上界为$\mathcal{O}(6^{n/3})$。此外，我们提供了一种分治算法，利用结构性质无需穷举即可精确计算可分离图的欧拉定向数量。最后，我们分析了用于从较小图生成$4$-正则图的标准归纳构造操作（对简单图见此http URL此http URL（1983），对多重图见此http URL此http URL（2003））对欧拉定向数量的影响。
英文摘要
An Eulerian orientation of a $4$-regular undirected graph (simple or multigraph) $G=(V,E)$ with $n=|V|$ vertices is an assignment of directions to its edges such that every vertex $v \in V$ has the same indegree and outdegree. In the present article, we improve the bounds on the number of Eulerian orientations for certain classes of connected, loopless $4$-regular graphs. The previous bound is due to M. Las Vergnas (1983) and is exactly $9\cdot 2^{n-3}$, which is a sharp bound for a certain family of multigraphs with $n\geq 4$. Here, we show that the number of Eulerian orientations for all biconnected $4$-regular multigraphs is at most $2^n+2$, which is also sharp. We exhibit families of graphs that attain this maximum value. For simple graphs, we prove an upper bound of $\mathcal{O}(3^{n/2})$ in the biconnected case and $\mathcal{O}(6^{n/3})$ for the separable case. Additionally, we provide a divide-and-conquer algorithm that leverages structural properties to compute the exact number of Eulerian orientations for separable graphs without exhaustive enumeration. Finally, we analyze the effect of standard inductive construction operations, used to generate $4$-regular graphs from smaller ones, as shown by F.Bories et.al. (1983) for simple graphs and by G.Ding et.al. (2003) for multigraphs, on the number of Eulerian orientations.
