The local connectivity \(k(u,v)\) between two distinct vertices \(u\) and \(v\) of a graph \(G\) is the maximum number of internally disjoint paths between \(u\) and \(v\) in \(G\). Let \(d(x)\) be the degree of a vertex \(x\). A graph \(G\) is maximally locally connected when \(k(u,v) = \min(d(u), d(v))\) for all pairs of \(u\) and \(v\) in \(G\). The author establishes a sufficient condition for equality of the connectivity and the minimum degree of a \(p\)-partite maximally locally connected graph.