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...

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 ≥ ...

Introduction. The line-graph L(G) of a nonempty graph G is the graph whose point set can be put in one-to-one correspondence with the line set of G in such a way that two points of L(G) are adjacent if and only if the corresponding lines of G are adjacent. In this paper graphs whose line-graphs are eulerian or hamiltonian are investigated and characterizations of these graphs are given. Further...

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, implies the existence of the (non-homogeneous) cd-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian po...

An Eulerian walk (or trail) is a (resp. that visits every edge of graph G at least exactly) once. This notion was first discussed by Leonhard Euler while solving the famous Seven Bridges Königsberg problem in 1736. But what if had to take bus? In temporal $$(G,\lambda )$$ , with $$\lambda : E(G)\rightarrow 2^{[\tau ]}$$ an $$e\in E(G)$$ available only times specified (e)\subseteq [\tau ]$$ same...

It has long been known that the number of spanning trees in circulant graphs with fixed jumps and n nodes satisfies a recurrence relation in n. The proof of this fact was algebraic (relating the products of eigenvalues of the graphs’ adjacency matrices) and not combinatorial. In this paper we derive a straightforward combinatorial proof of this fact. Instead of trying to decompose a large circu...