In [J Combin Theory Ser B, 26 (1979), 205–216], Jaeger showed that every graph with 2 edge-disjoint spanning trees admits a nowhere-zero 4-flow. In [J Combin Theory Ser B, 56 (1992), 165–182], Jaeger et al. extended this result by showing that, if A is an abelian group with |A| = 4, then every graph with 2 edgedisjoint spanning trees is A-connected. As graphs with 2 edge-disjoint spanning trees...

We answer a question on group connectivity suggested by Jaeger et al. [Group connectivity of graphs – A nonhomogeneous analogue of nowhere-zero flow properties, JCTB 1992]: we find that Z2-connectivity does not imply Z4-connectivity, neither vice versa. We use a computer to find the graphs certifying this and to verify their properties using nontrivial enumerative algorithm. While the graphs ar...

Tutte introduced the theory of nowhere zero flows and showed that a plane graph G has a face k-coloring if and only if G has a nowhere zero A-flow, for any Abelian group A with |A| ≥ k. In 1992, Jaeger et al. [9] extended nowhere zero flows to group connectivity of graphs: given an orientationD of a graph G, if for any b : V (G) → Awith  v∈V (G) b(v) = 0, there always exists a map f : E(G) →...

The existence of an integral flow polynomial that counts nowhere-zero k-flows on a graph, due to Kochol, is a consequence of a general theory of inside-out polytopes. The same holds for flows on signed graphs. We develop these theories, as well as the related counting theory of nowhere-zero flows on a signed graph with values in an abelian group of odd order. Our results are of two kinds: polyn...

Cai an Corneil (Discrete Math. 102 (1992) 103–106), proved that if a graph has a cycle double cover, then its line graph also has a cycle double cover, and that the validity of the cycle double cover conjecture on line graphs would imply the truth of the conjecture in general. In this note we investigate the conditions under which a graph G has a nowhere zero kow would imply that L(G), the line...

In this paper we will survey some recent results on the Hamiltonian dynamics of the geodesic flow of a Riemannian manifold. More specifically, we are interested in those manifolds which admit a Riemannian metric for which the geodesic flow is integrable. In Section 2, we introduce the necessary topics from symplectic geometry and Hamiltonian dynamics (and, in particular, we defined the terms ge...