The Crossing Number of Twisted Graphs

We consider twisted graphs , that is, topological graphs that are weakly isomorphic to subgraphs of the complete twisted graph T_n. We determine the exact minimum number of crossings of edges among the set of twisted graphs with n vertices and m edges; state a version of the crossing lemma for twisted graphs and conclude that the mid-range crossing constant for twisted graphs is 1/6. Let e_k(n) be the maximum number of edges over all twisted graphs with n vertices and local crossing number at most k . We give lower and upper bounds for e_k(n) and settle its exact value for k∈{0,1,2,3,6,10}. We conjecture that for every t≥ 1, e_( [ t; 2; ]) (n) = (t + 1)n - ( [ t + 2; 2; ]), n ≥ t+1.