THE MAXIMUM GENUS OF GRAPHS OF DIAMETER 2

Let G be a (finite) graph of diameter two. We prove that if G is loopless then it is upper embeddable, i.e. the maximum genus gamma-m(G) equals [beta(G)/2], where beta(G) = \E(G)\ - \V(G)\ + 1 is the Betti number of G. For graphs with loops we show that [beta(G)/2] - 2 less-than-or-equal-to gamma-M(G) less-than-or-equal-to [beta(G)/2] if G is vertex 2-connected, and compute the exact value of gamma-M(G) if the vertex-connectivity of G is 1. We note that by a result of Jungerman [2] and Xuong [10] 4-connected graphs are upper embeddable.