Bounds for the genus of graphs with given Betti number

Abstract We consider the problem of the relationship of the genus of a graph to its Betti number. We define the function g(β) which is the largest genus which occurs among graphs of Betti number β. Kuratowski showed that g(β) = 0 for β = 1, 2, 3 and that g(4) = 1. We review all known results for small β. Graphs of large girth are used to obtain lower bounds for g(β) for large β, and estimates of the number of disjoint circuits in a cubic graph give upper bounds: β 2 + 1 2 − 3(β−1) log 2 ( 3 2 β−1) ⩽g(β)⩽ β 2 − β 4log 2 β . This shows that the quantity β 2 − g(β) is bounded from above as well as below by const. β log β . We also give an improved estimate for the minimum number of nodes necessary for a cubic graph of given girth.