Cubic Graphs without a Petersen Minor Have Nowhere–zero 5–flows

We show that every bridgeless cubic graph without a Petersen minor has a nowhere-zero 5-flow. This approximates the known 4-flow conjecture of Tutte. A graph has a nowhere-zero k-flow if its edges can be oriented and assigned nonzero elements of the group Zk so that the sum of the incoming values equals the sum of the outcoming ones for every vertex of the graph. An equivalent definition we get using any Abelian group of order k or integers ±1, . . . , ±(k−1), as follows from Tutte [13], [14], [15] (see also [5], [16]). It is also known that graphs with bridges (1-edge-cuts) have no nowhere-zero k-flows for any k ≥ 2, and that if a graph has a nowhere-zero k-flow, then it has a nowhere-zero (k + 1)-flow. There are three celebrated conjectures dealing with nowhere-zero flows in bridgeless graphs, all due to Tutte. The first is the 5-flow conjecture of [13], that every such graph admits a nowhere-zero 5-flow. The 3-flow conjecture states that if the graph does not contain a 3-edge cut, then it has a nowhere-zero 3-flow. Finally the 4-flow conjecture of [15] suggests that if the graph does not contain a subgraph contractible to the Petersen graph, then it has a nowhere-zero 4-flow. The last conjecture has also a second variant, where are considered only cubic (3-regular) graphs. It is not known, whether the second variant implies the first one. We know the best possible approximations for the 5and 3-flow conjectures, because Seymour [12] and Jaeger [4] proved that every bridgeless and 4-edgeconnected graphs have nowhere-zero 6and 4-flows, respectively. We show that a similar approximation holds also for the cubic variant of the 4-flow conjecture, i.e., that every bridgeless cubic graph without a Petersen minor has a nowherezero 5-flow. The proof is an easy corollary of two already known results. The first one is a strong structural theorem announced recently by Robertson, Seymour and Thomas [10] (mentioned also in [16, Theorem 3.7.16]). Theorem 1. Every cubic graph without a Petersen minor has girth at most 5. Received January 10, 1999. 1980 Mathematics Subject Classification (1991 Revision). Primary 05C15.