Crossing numbers of sequences of graphs I: general tiles

A tile T is a connected graph together with two specified sequences of vertices, the left and right walls. The crossing number tcr(T ) of a tile T is the minimum number of crossings among all drawings of T in the unit square with the left wall in order down the left hand side and the right wall in order down the right hand side. The tile T n is obtained by gluing n copies of T in a linear fashion, while the graph ◦(T ) is obtained by gluing n copies of T in a circular fashion. Our main theorem is: limn→∞ tcr(T )/n = limn→∞ cr(◦(T ))/n. Thus, for any tile T , there are constants acr(T ) and cT such that n · acr(T ) − cT ≤ cr(◦(T )) ≤ n · acr(T ) + cT .