We show that the problem of counting the number of Eulerian circuits in an undirected graph is complete for the class #P. The method employed is mod-p reduction from counting Eulerian orientations.

Let G be a simple connected graph all of whose vertices have even degrees. An Eulerian circuit in G is a closed walk (see, for example, [2]) which uses every edge of G exactly once. Two Eulerian circuits are called equivalent if one is a cyclic permutation of the other. It is clear that the size of such an equivalence class equals the number of edges of graph G. Let EC(G) denote the number of e...

The (q, r)-Eulerian polynomials are the (maj−exc, fix, exc) enumerative polynomials of permutations. Using Shareshian and Wachs’ exponential generating function of these Eulerian polynomials, Chung and Graham proved two symmetrical q-Eulerian identities and asked for bijective proofs. We provide such proofs using Foata and Han’s three-variable statistic (inv−lec, pix, lec). We also prove a new ...

1.1. Eulerian and Lagrangian coordinates. Let us begin with Eulerian and Lagrangian coordinates. The Eulerian coordinate (x, t) is the physical space plus time. The Eulerian description of the flow is to describe the flow using quantities as a function of a spatial location x and time t, e.g. the flow velocity u(x, t). This can be visualized by sitting on the bank of a river and watching the wa...

A short report on my lecture at the Conference in Honor of W. Killing in Münster on December 7, 2007, together with some remarks on the preceding paper by ArjehM. Cohen. I reported on Euler’s famous paper Remarques sur un beau rapport entre les series des puissances tant direct que reciproques, Académie des sciences de Berlin, Lu en 1749, Opera Omnia Serie I, Bd. 15, pp. 70-90. In modern termin...

We show that each Eulerian representation of n is the restriction of a representation of n+1. We describe the new representations, giving character formulae, and identify the one which restricts to the rst Eulerian representation as the tree representation.

Three Eulerian models are implemented on a network graph model of air traffic flow. The underlying network is constructed using one year of ASDI/ETMS data. The three models are applied to high altitude traffic for a full Air Route Traffic Control Center of the National Airspace System and surrounding airspace. Simulations are carried out for a full day of data, to assess the predictive capabili...

Fleischner, H., G. Sabidussi and E. Wenger, Transforming eulerian trails, Discrete Mathematics 109 (1992) 103-116. In this paper a set of transformations (K-transformations) between eulerian trails is investigated. It is known that two arbitrary eulerian trails can be transformed into each other by a sequence of K-transformations. For compatible eulerran trails the set of K-transformations is a...

For a graph G = (V,E) and a vertex v ∈ V , let T (v) be a local trace at v, i.e. T (v) is an Eulerian subgraph of G such that every walk W (v), with start vertex v can be extended to an Eulerian tour in T (v). We prove that every maximum edge-disjoint cycle packing Z∗ of G induces a maximum trace T (v) at v for every v ∈ V . Moreover, if G is Eulerian then sufficient conditions are given that g...