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 γ(G)⩾g−22(g−1)β(G)+1g−1 except for G=K4, where β(G) denotes the cycle rank of G and K4 is the complete graph with four vertices.