Nowhere-zero 3-flows of graphs with prescribed sizes of odd edge cuts

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...

Cycles Intersecting Edge-Cuts of Prescribed Sizes

We prove that every cubic bridgeless graph G contains a 2-factor which intersects all (minimal) edge-cuts of size 3 or 4. This generalizes an earlier result of the authors, namely that such a 2-factor exists provided that G is planar. As a further extension, we show that every graph contains a cycle (a union of edge-disjoint circuits) that intersects all edge-cuts of size 3 or 4. Motivated by t...

Realizing Degree Sequences with Graphs Having Nowhere-Zero 3-Flows

The following open problem was proposed by Archdeacon: Characterize all graphical sequences π such that some realization of π admits a nowhere-zero 3-flow. This open problem is solved in this paper with the following complete characterization: A graphical sequence π = (d1, d2, . . . , dn) with minimum degree at least two has a realization that admits a nowhere-zero 3-flow if and only if π 6= (3...

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