On the Resolution Complexity of Graph Non-isomorphism

For a pair of given graphs we encode the isomorphism principle in the natural way as a CNF formula of polynomial size in the number of vertices, which is satisfiable if and only if the graphs are isomorphic. Using the CFI graphs from [12], we can transform any undirected graph G into a pair of non-isomorphic graphs. We prove that the resolution width of any refutation of the formula stating that these graphs are isomorphic has a lower bound related to the expansion properties of G. Using this fact, we provide an explicit family of non-isomorphic graph pairs for which any resolution refutation requires an exponential number of clauses in the size of the initial formula. These graphs pairs are colored with color multiplicity bounded by 4. In contrast we show that when the color classes are restricted to have size 3 or less, the nonisomorphism formulas have tree-like resolution refutations of polynomial size.