Refined Bounds on the Number of Eulerian Tours in Undirected Graphs

Abstract

Given an undirected multigraph $G=(V,E)$ with no self-loops, and one of its nodes $s\in V$, we consider the #P-complete problem of counting the number $ET^{(e)}_s(G)$ of its Eulerian tours starting and ending at node s. We provide lower and upper bounds on the size of $ET^{(e)}_s(G)$. Namely, let d(v) denote the degree of a node $v\in V$; we show that $\max \{L_1^{(e)}, L_2^{(e)}\} \le |ET^{(e)}_{s}(G)| \le d(s)\, \prod _{v \in V} (d(v) - 1)!!$ where $L_1^{(e)} = (d(s)-1)!!\prod _{v \in V {\setminus } s}{(d(v)-2)!!}$ and $L_2^{(e)} = 2^{1-|V|+|E|}$. We also consider the notion of node-distinct Eulerian tours. Indeed, the classical Eulerian tours are edge-distinct sequences. Node-distinct Eulerian tours, denoted $ET^{(n)}_s(G)$, should instead be different as node sequences. Let $\Delta (u)$ be the number of distinct neighbors of a node u, $D \subseteq E$ be the set of distinct edges in the multigraph G, and m(e) for an edge $e\in E$ be its multiplicity (i.e. $|E|=\sum _{e \in D} m(e)$). We prove that $\max \{L_1^{(n)}, L_2^{(n)}, L_3^{(n)}\} \le |ET^{(n)}_{s}(G)| \le d(s)\, \prod _{v \in V} (d(v) - 1)!! \cdot \textstyle \prod _{e\in D} m(e)!^{-1},$ where $L_1^{(n)} = L_1^{(e)}/(\prod _{e \in D}m(e)!)$, $L_2^{(n)} = (\Delta (s)-1)!!\prod _{v \in V {\setminus } s}{(\Delta (v)-2)!!}$, and $L_3^{(n)} = 2^{1-|V|+|D|}$. We also extend all of our results to graphs having self-loops.