Colorings undirected G(V,E) max-degree: q colors: d 口 temporal mixing of approximately counting Glauber dynamics or sampling almost uniform 0:2→11/6 proper q-colorings of G [Jerrum'95] [Vigoda'99] [Salas-Sokal'97] [Bubley-Dyer'97] when q≥cd+B spatial mixing of conjecture:x=1 Gibbs measureColorings undirected G(V,E) approximately counting or sampling almost uniform proper q-colorings of G max-degree: q colors: d when q ≥αd+ temporal mixing of Glauber dynamics β spatial mixing of Gibbs measure ? α: 2 → 11/6 [Jerrum’95] [Salas-Sokal’97] [Bubley-Dyer’97] [Vigoda’99] conjecture: α=1