Let G be a graph. For each vertex v 2V (G), Nv denotes the subgraph induces by the vertices adjacent to v inG. The graphG is locally k edge-connected if for each vertex v 2V (G), Nv is k-edge-connected. In this paper we study the existence of nowhere-zero 3-flows in locally k-edgeconnected graphs. In particular, we show that every 2-edge-connected, locally 3-edge-connected graph admits a nowher...

Let $$\varGamma $$ be a graph, A an abelian group, $${\mathcal {D}}$$ given orientation of and R unital subring the endomorphism ring A. It is shown that set all mappings $$\varphi from $$E(\varGamma )$$ to such $$({\mathcal {D}},\varphi A-flow forms left R-module. union two subgraphs _{1}$$ _{2}$$ , $$p^n$$ prime power. proved admits nowhere-zero -flow if have at most $$p^n-2$$ common edges bo...

In 1972, Tutte posed the 3-Flow Conjecture: that all 4-edge-connected graphs have a nowhere-zero 3-flow. This was extended by Jaeger et al. to allow vertices prescribed, possibly nonzero difference (modulo 3) between inflow and outflow. They conjectured 5-edge-connected with prescription function 3-flow meeting prescription. Kochol showed replacing would suffice prove Conjecture Lovász both con...

In [Discrete Math. 230 (2001), 133-141], it is shown that Tutte’s 3-flow conjecture that every 4-edge-connected graph has a nowhere zero 3-flow is equivalent to that every 4-edge-connected line graph has a nowhere zero 3-flow. We prove that every line graph of a 4-edgeconnected graph is Z3-connected. In particular, every line graph of a 4-edge-connected graph has a nowhere zero 3-flow.

Tension-continuous (shortly TT ) mappings are mappings between the edge sets of graphs. They generalize graph homomorphisms. From another perspective, tension-continuous mappings are dual to the notion of flow-continuous mappings and the context of nowhere-zero flows motivates several questions considered in this paper. Extending our earlier research we define new constructions and operations f...

In 1972, Tutte posed the $3$-Flow Conjecture: that all $4$-edge-connected graphs have a nowhere zero $3$-flow. This was extended by Jaeger et al.(1992) to allow vertices prescribed, possibly non-zero difference (modulo $3$) between inflow and outflow. They conjectured $5$-edge-connected with valid prescription function $3$-flow meeting (we call this Strong Conjecture). Kochol (2001) showed repl...