On Nowhere Zero 4-Flows in Regular Matroids

Walton and Welsh proved that if a co-loopless regular matroid M does not have a minor in {M(K(3,3)),M∗(K5)}, then M admits a nowhere zero 4-flow. Lai, Li and Poon proved that if M does not have a minor in {M(K5),M∗(K5)}, then M admits a nowhere zero 4-flow. We prove that if a co-loopless regular matroid M does not have a minor in {M((P10)¯3 ),M∗(K5)}, then M admits a nowhere zero 4-flow where (P10)¯3 is the graph obtained from the Petersen graph P10by contracting 3 edges of a perfect matching. As both M(K3,3) and M(K5) are contractions of M((P10)¯3), our result extends the results of Walton and Welsh and Lai, Li and Poon.