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 ( P 10 ) μ ̄ be the graph obtained from the Petersen graph by contracting μ edges from a perfect matching. In this paper we prove that every bridgeless ( P 10 ) 3 ̄ -minor free graph admits a nowhere-zero 4-flow.