The maximum genus of graphs of diameter two

Skoviera, M., The maximum genus of graphs of diameter two, Discrete Mathematics 87 (1991) 175-180. 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 y,&G) equals [fi(G)/Z], where /3(G) = IF(G)1 IV(G)1 + 1 is the Betti number of G. For graphs with loops we show that [p(G)/21 2s yM(G) c &G)/Z] if G is vertex 2-connected, and compute the exact value of yM(G) if the vertex-connectivity of G is 1. We note that by a result of Jungerman [2] and Xuong [lo] 4-connected graphs are upper embeddable. Introduction aud statement of main results This paper is devoted to an investigation of the maximum genus of graphs of diameter two with multiple adjacencies and loops permitted. Unlike the diameter, the maximum genus is invariant under homeomorphisms therefore the results presented below obviously extend to graphs homeomorphic to those of diameter two. Recall that the maximum genus yM(G) of a connected graph G is the largest genus of an orientable surface on which G has a 2-cell embedding. (For basic information and results see [l, Section 5.31 and [7].) Leshchenko [4] proved that every simple graph of diameter two with even Betti number admits a 2-cell embedding on an orientable surface with one region. In the current terminology, he has shown that such graphs are upper embeddable. It seems to be worth asking whether all (simple) graphs of diameter two are upper embeddable. Our Theorem 1 presents an affirmative answer to this question, which completes and slightly generalizes the result of Leshchenko by allowing multiple edges. Theorem 1. Every loopless graph of diameter two is upper embeddable. 0012-365X/91/$03.50