Group Connectivity and Matchings

Abstract

Let $$\alpha ^\prime (G)$$??(G) be the maximum number of independent edges in a graph $$G$$G and $$A$$A an abelian group with $$|A|\ge 4$$|A|?4. If a graph $$G^*$$G? is obtained by repeatedly contracting nontrivial $$A$$A-connected subgraphs of $$G$$G until no such a subgraph left, then we say $$G$$G can be $$A$$A-reduced to $$G^*$$G?. In this paper, we show that if $$G$$G is a 3-edge-connected simple graph and $$\alpha ^\prime (G)\le 5$$??(G)≤5, then either $$G$$G is $$A$$A-connected or $$G$$G can be $$A$$A-reduced to the Petersen graph.