  We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the
Ollivier Ricci curvature sense. Our main result is a classification of all
self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs,
even-dimensional demi-cubes, Johnson graphs $J(2n,n)$, the Gosset graph and
suitable Cartesian products). We also present a purely combinatorial
reformulation of this result. We show that Bonnet-Myers sharpness implies
Lichnerowicz sharpness. We also relate Bonnet-Myers sharpness to an upper bound
of Bakry-\'Emery $\infty$-curvature, which motivates a generalconjecture about
Bakry-\'Emery $\infty$-curvature.
