On local connectivity of K-2,K-p-free graphs

For a vertex v of a graph G, we denote by d(v) the degree of v. The local connectivity kappa(u, v) of two vertices u and v in a graph G is the maximvm nvmber of internally disjoint u-v paths in G. Clearly, kappa(u, v) <= min{d(u), d(v)} for all pairs u and v of vertices in G. We call a graph G maximally local connected when kappa(u, v) = min{d(u), d(v)} for all pairs u and v of distinct vertices in G. Let p >= 2 be an integer. We call a graph K-2,(p)-free if it contains no complete bipartite graph K-2,K-p as a (not necessarily indvced) svbgraph. If p >= 3 and G is a connected K-2,(p)-free graph of order n and minimvm degree delta svch that n <= 3 delta-2p+2, then G is maximally local connected dve to ovr earlier resvlt on p-diamond-free graphs [Discrete Math. 309 (2009), 6065-6069]. Now we present examples showing that the condition n <= 3 delta-2p+2 is best possible for p = 3 and p >= 5. In the case p = 4 we present the improved condition n <= 3 delta-5 implying maximally local connectivity. In addition, we present similar resvlts for K-2,K-2-free graphs.