Sparse cycle bases for graphs with bounded genus

The basis number of a graph G is the smallest integer k such that G admits a basis B for its cycle space, where each edge of G belongs to at most k members of B . In this note, we show that every non-planar graph that can be embedded on a surface with Euler characteristic 0 has a basis number of exactly 3, proving a conjecture of Schmeichel from 1981. Additionally, we show that any graph embeddable on a surface Σ (whether orientable or non-orientable) of genus g has a basis number of O ( log ( g ) 2 ) .