Join of two graphs admits a nowhere-zero 3-flow

Nowhere-Zero 3-Flows in Signed Graphs

Tutte observed that every nowhere-zero k-flow on a plane graph gives rise to a kvertex-coloring of its dual, and vice versa. Thus nowhere-zero integer flow and graph coloring can be viewed as dual concepts. Jaeger further shows that if a graph G has a face-k-colorable 2-cell embedding in some orientable surface, then it has a nowhere-zero k-flow. However, if the surface is nonorientable, then a...

Nowhere-zero flow polynomials

In this article we introduce the flow polynomial of a digraph and use it to study nowherezero flows from a commutative algebraic perspective. Using Hilbert’s Nullstellensatz, we establish a relation between nowhere-zero flows and dual flows. For planar graphs this gives a relation between nowhere-zero flows and flows of their planar duals. It also yields an appealing proof that every bridgeless...

Nowhere-Zero 3-Flows in Squares of Graphs

It was conjectured by Tutte that every 4-edge-connected graph admits a nowherezero 3-flow. In this paper, we give a complete characterization of graphs whose squares admit nowhere-zero 3-flows and thus confirm Tutte’s 3-flow conjecture for the family of squares of graphs.

Nowhere-zero 3-flows in Cayley Graphs of Abelian Groups

Nowhere-zero 3-flows in products of graphs

A graph G is an odd-circuit tree if every block of G is an odd length circuit. It is proved in this paper that the product of every pair of graphs G and H admits a nowhere-zero 3-flow unless G is an odd-circuit tree and H has a bridge. This theorem is a partial result to the Tutte’s 3-flow conjecture and generalizes a result by Imrich and Skrekovski [7] that the product of two bipartite graphs ...