Nowhere-zero 3-flows of claw-free graphs

Let A denote an abelian group and G be a graph. If a graph G ∗ is obtained by repeatedly contracting nontrivial A -connected subgraphs of G until no such a subgraph left, we say G can be A -reduced to G ∗ . A graph is claw-free if it has no induced subgraph K 1 , 3 . Let N 1 , 1 , 0 denote the graph obtained from a triangle by adding two edges at two distinct vertices of the triangle, respectively. In this paper, we prove that if G is a simple 2-connected { claw , N 1 , 1 , 0 } -free graph, then G does not admit nowhere-zero 3-flow if and only if G can be Z 3 -reduced to two families of well characterized graphs or G is one of the five specified graphs.