Nowhere-zero 4-flow in almost Petersen-minor free graphs

Tutte [W.T. Tutte, On the algebraic theory of graph colorings, J. Combin. Theory 1 (1966) 15–20] conjectured that every bridgeless Petersen-minor free graph admits a nowhere-zero 4-flow. Let (P10)μ̄ be the graph obtained from the Petersen graph by contracting μμ edges from a perfect matching. In this paper we prove that every bridgeless (P10)3̄-minor free graph admits a nowhere-zero 4-flow.