The b-chromatic number of regular graphs via the edge-connectivity

A d-regular graph G is super-edge-connected if every minimum edge-cut is the set of all edges incident to one vertex in G, i.e., the edge-connectivity of G is equal to d and deleting every minimum edge-cut of G isolates a vertex. We show that if G is a d-regular graph that contains no 4-cycle as a subgraph, then φ(G)=d+1 whenever G is not super-edge-connected.