Not every graph has an Eulerian tour. But every finite, strongly connected graph has a multi-Eulerian tour, which we define as a closed path that uses each directed edge at least once, and uses edges e and f the same number of times whenever tail(e) = tail(f). This definition leads to a simple generalization of the BEST Theorem. We then show that the minimal length of a multi-Eulerian tour is b...

In this paper we obtain the expectation and variance of the number of Euler tours of a random Eulerian directed graph with fixed out-degree sequence. We use this to obtain the asymptotic distribution of the number of Euler tours of a random d-in/d-out graph and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler...

let $g$ be a simple graph with an orientation $sigma$‎, ‎which ‎assigns to each edge a direction so that $g^sigma$ becomes a‎ ‎directed graph‎. ‎$g$ is said to be the underlying graph of the‎ ‎directed graph $g^sigma$‎. ‎in this paper‎, ‎we define a weighted skew‎ ‎adjacency matrix with rand'c weight‎, ‎the skew randi'c matrix ${bf‎ ‎r_s}(g^sigma)$‎, ‎of $g^sigma$ as the real skew symmetric mat...

For an integer s > 0 and for u, v ∈ V (G) with u ≠ v, an (s; u, v)-trail-system of G is a subgraphH consisting of s edge-disjoint (u, v)-trails. A graph is supereulerianwithwidth s if for any u, v ∈ V (G) with u ≠ v, G has a spanning (s; u, v)-trail-system. The supereulerian width μ(G) of a graph G is the largest integer s such that G is supereulerian with width k for every integer kwith 0 ≤ k ...

This paper defines the Stochastic Eulerian Tour Problem (SETP) and investigates several characteristics of this problem. Given an undirected Eulerian graph , a subset =) of the edges in E that require service, and a probability distribution for the number of edges in R that have to be visited in any given instance of the graph, the SETP seeks an a priori Eulerian tour of minimum expected length...