The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G ... more The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [B. Richter, C. Thomassen, Minimal graphs with crossing number at least k, J. Combin. Theory Ser. B 58 (1993) 217-224] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that cr(G − e) k − c √ k. They showed only that G always has an edge e with cr(G − e) 2 5 cr(G) − O(1). We prove that for every fixed > 0, there is a constant n 0 depending on such that if G is a graph with n > n 0 vertices and m > n 1+ edges, then G has a subgraph G with at most (1 − 24 )m edges such that cr(G ) ( 1 28 − o(1)) cr(G).
The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G ... more The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [RT93] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that cr(G − e) ≥ k − c √ k. They showed only that G always has an edge e with cr(G − e) ≥ 2 5 cr(G) − O(1). We prove that for every fixed > 0, there is a constant n0 depending on such that if G is a graph with n > n0 vertices and m > n 1+ edges, then G has a subgraph G with at most (1 − 1 24 )m edges such that cr(G ) ≥ ( 1 28 − o(1))cr(G).
The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G ... more The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [B. Richter, C. Thomassen, Minimal graphs with crossing number at least k, J. Combin. Theory Ser. B 58 (1993) 217-224] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that cr(G − e) k − c √ k. They showed only that G always has an edge e with cr(G − e) 2 5 cr(G) − O(1). We prove that for every fixed > 0, there is a constant n 0 depending on such that if G is a graph with n > n 0 vertices and m > n 1+ edges, then G has a subgraph G with at most (1 − 24 )m edges such that cr(G ) ( 1 28 − o(1)) cr(G).
The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G ... more The crossing number cr(G) of a graph G is the minimum number of crossings over all drawings of G in the plane. In 1993, Richter and Thomassen [RT93] conjectured that there is a constant c such that every graph G with crossing number k has an edge e such that cr(G − e) ≥ k − c √ k. They showed only that G always has an edge e with cr(G − e) ≥ 2 5 cr(G) − O(1). We prove that for every fixed > 0, there is a constant n0 depending on such that if G is a graph with n > n0 vertices and m > n 1+ edges, then G has a subgraph G with at most (1 − 1 24 )m edges such that cr(G ) ≥ ( 1 28 − o(1))cr(G).
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