Extending a partial nowhere-zero 4-flow

In [J Combin Theory Ser B, 26 (1979), 205–216], Jaeger showed that every graph with 2 edge-disjoint spanning trees admits a nowhere-zero 4-flow. In [J Combin Theory Ser B, 56 (1992), 165–182], Jaeger et al. extended this result by showing that, if A is an abelian group with |A| = 4, then every graph with 2 edgedisjoint spanning trees is A-connected. As graphs with 2 edge-disjoint spanning trees are all collapsible, we in this note improve the latter result by showing that, if A is an abelian group with |A| = 4, then every collapsible graph is A-connected. This allows us to prove the following generalization of Jaeger’s theorem: Let G be a graph with 2 edge-disjoint spanning trees and let M be an edge cut of G with |M | ≤ 4. Then either any partial nowhere-zero 4-flow on M can be extended to a nowhere-zero 4-flow of the whole graph G, or G can be contracted to one of three configurations, including the wheel of 5 vertices, in which cases certain partial nowhere-zero 4-flows on M cannot be extended. Our results also improve a theorem of Catlin in [J Graph Theory, 13 (1989), 465–483]. c © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 277–288, 1999