Nowhere-zero 3-flows and Z3Z3-connectivity of graphs without two forbidden subgraphs

Jaeger et al.’s Z3Z3-connectivity conjecture can be reduced to a consideration of 5-edge-connected K1,3K1,3-free graphs by Lovász et al. (2013) and Ma et al. (2014). Let K1,3+ denote the graph obtained from K1,3K1,3 by adding an edge connecting two vertices of degree 1. Denote by K1,3∗ the graph obtained from a K1,3+ by adding an edge to one vertex of degree 1. In this paper, we will prove the following two results.(1) If GG is a 2-connected {K1,3,K1,3+}-free simple graph, then GG is Z3Z3-connected if and only if GG is not one of K4K4, K4− or an nn-cycle, where n≥3n≥3.(2) If GG is a 2-connected {K1,3,K1,3∗}-free simple graph, then GG is not Z3Z3-connected if and only if GG is isomorphic to one of the 20 specified graphs or GG is an nn-cycle, where n≥3n≥3.