Theory and Applications of Graphs Theory and Applications of Graphs

Abstract

Walton and Welsh proved that if a coloopless regular matroid M does not have a minor in { M ( K 3 , 3 ) , M ∗ ( K 5 ) } , then M admits a nowhere zero 4-flow. Lai, Li and Poon proved that if M does not have a minor in { M ( K 5 ) , M ∗ ( K 5 ) } , then M admits a nowhere zero 4-flow. We prove that if a coloopless regular matroid M does not have a minor in { M (( P 10 ) ¯3 ) , M ∗ ( K 5 ) } , then M admits a nowhere zero 4-flow where ( P 10 ) ¯3 is the graph obtained from the Petersen graph P 10 by contracting 3 edges of a perfect matching. As both M ( K 3 , 3 ) and M ( K 5 ) are contractions of M (( P 10 ) ¯3 ), our result extends the results of Walton and Welsh and Lai, Li and Poon.