Let \(\mathcal {B}\) be a set of Eulerian subgraphs of a graph G. We say \(\mathcal {B}\) forms a if it is a minimum set that generates the cycle space of G, and any edge of G lies in at most k members of \(\mathcal {B}\) . The of a graph G, denoted by b(G), is the smallest integer such that G has a k-basis. A graph is called (resp. ) if it can be embedded in the plane with at most one crossing (resp. no crossing) per edge. MacLane’s planarity criterion characterizes planar graphs based on their cycle space, stating that a graph is planar if and only if it has a 2-basis. We study here the basis number of 1-planar graphs, demonstrate that it is unbounded in general, and show that it is bounded for many subclasses of 1-planar graphs.
