On the lengths of symmetry breaking–preserving games on graphs

Abstract

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 L sym( 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)⩽L sym (G)⩽ O( log 2 n) . The lower bound is based on relations with Ehrenfeucht–Fraı̈ssé games from model theory. We also consider complete graphs and prove that L sym( K n )=O(1).