Addible edges in 2-matching-connected graphs and 3-matching-connected graphs

Abstract

Let k be a positive integer. A graph G is k-matching-connected if G−V(M) is connected and nontrivial, for each matching M of G with |M|≤k−1. For a k-matching-connected graph G, an edge e∉E(G) is said to be addible if G+e is still k-matching-connected. In this paper, we prove that every 2-matching-connected graph G has no addible edge if and only if G is a cycle of length at least 4. We also show that every 3-matching-connected graph G with δ(G)≥5 has an addible edge, and some examples are given to show that δ(G)=5 is the best possible lower bound.