The maximum genus of a connected graph G, γM(G), is the largest genus of an orientable surface on which G has a 2-cell embedding, and the Betti deficiency, ζ(G), is equal to β(G) - 2γM(G) where β(G) - |E(G)| - |V(G)| + 1 is the Betti number of G. In this paper we study the maximum genus of a graph with diameter k ≥ 3 and we prove that the Betti deficiency of a diameter 3 multigraph is at most 2. In the case that the diameter 3 graph G is simple, the Betti deficiency of G can be determined. As to graphs with larger diameter, some partial results are obtained.
|Number of pages||11|
|Journal||Australasian Journal of Combinatorics|
|State||Published - 1 Dec 1996|