Parity Subgraphs with Few Common Edges and Nowhere-Zero 5-Flow

Abstract

A parity subgraph of a graph is a spanning subgraph such that the degrees of all vertices have the same parity in both the subgraph and the original graph. Let G be a cyclically 6-edge-connected cubic graph. Steffen (Intersecting 1-factors and nowhere-zero 5-flows 1306.5645, 2013) proved that G has a nowhere-zero 5-flow if G has two perfect matchings with at most two intersections. In this paper, we show that G has a nowhere-zero 5-flow if G has two parity subgraphs with at most two common edges, which generalizes Steffen's result.