Nowhere-zero 3-flows of claw-free graphs

Abstract

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 K1,3. Let N1,1,0 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,N1,1,0}-free graph, then G does not admit nowhere-zero 3-flow if and only if G can be Z3-reduced to two families of well characterized graphs or G is one of the five specified graphs.