Nowhere-zero 3-flows and Z3-connectivity of graphs without two forbidden subgraphs

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...

Contractible configurations, Z3-connectivity, Z3-flows and triangularly connected graphs

Tutte conjectured that every 4-edge connected graph admits a nowhere-zero Z3-flow and Jaeger, Linial, Payan and Tarsi conjectured that every 5-edge connected graph is Z3-connected. In this paper, we characterize the triangularly connected graphs G that are Γ-connected for any Abelian group Γ with |Γ| ≥ 3. Therefore, these two conjectures are verified for the family of triangularly connected gra...

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 ...