Separating and Nonseparating Disjoint Homotopic Cycles in Graph Embeddings

We show that if a graphGis embedded in a surfaceΣwith representativityρ, thenGcontains at least ⌊(ρ−1)/2⌋ pairwise disjoint, pairwise homotopic, nonseparating (inΣ) cycles, andGcontains at least ⌊(ρ−1)/8⌋−1 pairwise disjoint, pairwise homotopic, separating, noncontractible cycles.