Number of Variables for Graph Differentiation and the Resolution of Graph Isomorphism Formulas

We show that the number of variables and quantifier depth needed to distinguish a pair graphs by first-order logic sentences exactly match complexity measures clause width refute corresponding graph isomorphism formula in propositional narrow resolution. Using this connection, we obtain upper lower bounds for refuting formulas (normal) In particular, if k is minimum two with n vertices each, then there an O ( ) resolution refutation size bound formula, as well 2 -1 treelike space formula. also exp (Ω / )) case colored constant color class sizes. Applying these results, prove first exponential proof system SRC-1, extends global symmetry rule, thereby answering open question posed Schweitzer Seebach.