The number of planar Eulerian maps with n edges is well-known to have a simple expression. But what is the number of planar Eulerian orientations with n edges? This problem appears to be difficult. To approach it, we define and count families of subsets and supersets of planar Eulerian orientations, indexed by an integer k, that converge to the set of all planar Eulerian orientations as k incre...

Let G be an eulerian digraph with a fixed edge coloring (not necessarily a proper edge coloring). A compatible circuit of G is an eulerian circuit such that every two consecutive edges in the circuit have different colors. We characterize the existence of compatible circuits for directed graphs avoiding certain vertices of outdegree three. Our result is analogous to a result of Kotzig for compa...

A graph G is Eulerian-connected if for any u and v in V (G), G has a spanning (u. v)-trail. A graph G is edge-Eulerian-connected if for any e' and elf in E(G), G has a spanning (e', elf)-trail. For an integer r ~ 0, a graph is called r-Eulerian-connected if for any X s::: E(G) with IXI ~r, and for any u, v E V(G), G has a spanning (u, v)-trail T such that X s::: E(T). The r-edge-Eulerian­ conne...

Using the existence of noncrossing Eulerian circuits in Eulerian plane graphs, we give a short constructive proof of the theorem of Heawood that Eulerian triangulations are 3-colorable.

Kim et al. 2012 introduced an interesting p-adic analogue of the Eulerian polynomials. They studied some identities on the Eulerian polynomials in connection with the Genocchi, Euler, and tangent numbers. In this paper, by applying the symmetry of the fermionic p-adic q-integral on Zp, defined by Kim 2008 , we show a symmetric relation between the q-extension of the alternating sum of integer p...

d-spheres are defined graph theoretically and inductively as the empty graph in dimension d = −1 and d-dimensional graphs for which all unit spheres S(x) are (d−1)-spheres and such that for d ≥ 0 the removal of one vertex renders the graph contractible. Eulerian d-spheres are geometric d-spheres which can be colored with d + 1 colors. They are Eulerian graphs in the classical sense and for d ≥ ...