Strong spatial mixing of $q$-colorings on Bethe lattices
  We investigate the problem of strong spatial mixing of $q$-colorings on Bethe lattices. By analyzing the sum-product algorithm we establish the strong spatial mixing of $q$-colorings on $(b+1)$-regular Bethe lattices, for $q \geq 1+\lceil 1.764b \rceil$. We also establish the strong spatial mixing of $q$-colorings on binary trees, for $q=4$. 
Qi Ge and Daniel Stefankovic
math.PR