Maximal local edge-connectivity of diamond-free graphs

The edge-connectivity of a graph G can be defined as λ(G) = min{λG(u, v) | u, v ∈ V (G)}, where λG(u, v) is the local edge-connectivity of two vertices u and v in G. We call a graph G maximally edge-connected when λ(G) =δ(G) and maximally local edge-connected when λG(u, v) =min{d(u),d(v)} for all pairs u and v of distinct vertices in G. In 2000, Fricke, Oellermann and Swart (unpublished manuscript) proved that a bipartite graph G of order n(G) is maximally local edge-connected when n(G) ≤ 4δ(G) − 1. As an extension of this result, we will show in this work that it is sufficient for G to be diamond-free with n(G) ≤ 4δ(G) − 1 to guarantee the maximally local edge-connectivity