Bounded-depth frege complexity of tseitin formulas for all graphs

We prove that there is a constant K such that Tseitin formulas for an undirected graph G requires proofs of size 2tw(G)Ω(1/d) in depth-d Frege systems for d < (Formula presented.) where tw(G) is the treewidth of G. This extends Håstad recent lower bound for the grid graph to any graph. Furthermore, we prove tightness of our bound up to a multiplicative constant in the top exponent. Namely, we show that if a Tseitin formula for a graph G has size s, then for all large enough d, it has a depth-d Frege proof of size 2tw(G)O(1/d)poly(s). Through this result we settle the question posed by M. Alekhnovich and A. Razborov of showing that the class of Tseitin formulas is quasi-automatizable for resolution.