Originators: ??? (presented by Gexin Yu - REGS 2007)
Definitions: Let A be an additive abelian group with identity element 0. A nowhere-zero A-flow on an oriented graph G is a weighting of the edge set by nonzero elements of A such that at each vertex the total flow in equals the total flow out. A graph is A-flowable if some orientation admits a nowhere-zero A-flow. It is k-flowable, where k is a positive integer, if it is A-flowable when A is the cyclic group of order k.
A graph G is A-connected if for every map c: V(G) → A with ∑ c(v) = 0, there is a nonzero weighting of an orientation of the edges of G such that at each vertex v, the total out-flow minus the total in-flow is c(v).
Background: The famous flow conjectures of Tutte are that every 2-edge-connected graph is 5-flowable, every 2-edge-connected graph with no Petersen minor is 4-flowable, and every 4-edge-connected graph is 3-flowable. If H is a subgraph of G, H is A-connected, and the graph obtained from G by contracting H to one vertex is A-flowable, then G is A-flowable [ref?]. Thomassen proved that every A-connected graph decomposes into claws. Luo, Xu, Yin, and Yu proved that Ore's Condition (the degrees of any two nonadjacent vertices sum to at least |V(G)|) implies that G is Z3-connected, with 12 exceptions. Fan & (??) proved that Ore's Condition implies that G is 3-flowable, with 6 exceptions.
Question 1: Let σk(G) denote the minimum sum of the degrees of the vertices in an independent k-set in G. The condition σk(G) ≥ |V(G)| is weaker than Ore's Condition, but strong enought to imply that G is Hamiltonian. Does it also imply that G is Z3-connected?
Conjecture (Luo): If an n-vertex graph G has average degree less than 3-(4/n), then G is not Z4-connected. (Avg deg less than 8/3 is sufficient)
Conjecture (Jaeger [1992]): A graph is Z4-connected if and only if it is (Z2×Z2)-connected.