Maximum genus and girth of graphs

In this paper, a lower bound on the maximum genus of a graph in terms of its girth is established as follows: let G be a simple graph with minimum degree at least three, and let g be the girth of G. Then gamma(M)(G) >= g-2/2(g-1) beta(G) + 1/g-1 except for G = K-4 , where beta(G) denotes the cycle rank of G and K-4 is the complete graph with four vertices. (C) 1999 Elsevier Science B.V. All rights reserved.