Computing and Bounding the Number of Eulerian Orientations for Certain Classes of $4$-Regular Graphs
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.