On the problems of CF-connected graphs

The crossing number cr( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane, and the optimal drawing of G is any drawing at which the desired minimum number of crossings is achieved. We conjecture that a complete graph K n is CF -connected if and only if it does not contain a subgraph of K 8 , where a connected graph G is CF -connected if there is a path between every pair of vertices with no crossing on its edges for each optimal drawing of G . We establish the validity of this Conjecture for the complete graphs K n for any n ≤ 12 , and by assuming the Harary-Hill’s Conjecture that cr( K n ) = H ( n ) = 14 (cid:4) n 2 (cid:5)(cid:4) n − 1 2 (cid:5)(cid:4) n − 2 2 (cid:5)(cid:4) n − 3 2 (cid:5) is also valid for all n > 12 . The proofs of this paper are based on the idea of a new concept of a crossing sequence.