Nowhere-zero3-flows And Z₃-connected Of K_1,3-free Graphs

Let H1and H2be two subgraphs of a graph G. If|V(H1)∩V(H2)|=1, E(H1)∩E(H2)=0and G=H1∪H2, then G is called1-sum of H1and H2and denote by G=H1(?) H2.For m≥3, define Fm=K4(?)Km, Tm=K3(?)Km, where Km is a complete graph on m vertices. It is proved in this paper that if G is2-edge-connected{K1,3,P4}-free simple graph, then G is not Z3-connected if and only if G is either one of the8specified graphs or one of Fm and Tm, where m≥3. As a corollary, if G is2-edge-connected{K1,3,P4}-free simple graph, then G admits no nowhere-zero3-flow if and only if G is one of the4specified graphs or one of Fm, where m≥3.