On the lengths of symmetry breaking-preserving games on graphs

Given a graph G, we consider a game where two players, A and B, alternatingly color edges of G in red and in blue respectively. Let Lsym(G) be the maximum number of moves in which B is able to keep the red and the blue subgraphs isomorphic, if A plays optimally to destroy the isomorphism. This value is a lower bound for the duration of any avoidance game on G under the assumption that B plays optimally. We prove that if G is a path or a cycle of odd length n, then Ω(log n) ≤ Lsym(G) ≤ O(log 2 n). The lower bound is based on relations with Ehrenfeucht-Fräıssé games from model theory. We also consider complete graphs and prove that Lsym(Kn) = O(1).