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 graphs. Let P be a graph theory property. A graph H is a P -contractible if, for every supgraph G of H (i.e. G is a graph containing H as a subgraph), G has the property P if and only if G/H has the property P . This concept is inspired by the following methods and techniques introduced by Catlin, Seymour, Jaeger, Linial, Payan and Tarsi, such as, collapsible graph for supereulerian graphs, Φk-graph in the proof of 6-flow theorem, and group connectivity for integer flows. In this paper, we proved some basic and useful lemmas for P -contractibility: (1) If a graph H is ∗Partially supported by the National Security Agency under Grants MDA90401-1-0022 and MSPR-03G-023.