A graph G is quasi 4-connected if it is simple, 3-connected, has at least five
vertices, and for every partition (A, B, C) of V(G) either
[mid ]C[mid ] [ges ] 4, or G has an edge with one end in A
and the other end in B, or one of A,B has at most one vertex. We show that any quasi
4-connected nonplanar graph with minimum degree at least three and no cycle of length
less than five has a minor isomorphic to P−10, the Petersen graph with one edge deleted. We
deduce the following weakening of Tutte's Four Flow Conjecture: every 2-edge-connected
graph with no minor isomorphic to P−10 has a nowhere-zero 4-flow. This extends a result
of Kilakos and Shepherd who proved the same for 3-regular graphs.