△-regular tree each v: proposes a uniform random color o,E[g]; update Xy to oy if for all y's neighbors u: X≠GvAG计XAGu≠Ov; Xroot red,Yroot blue V non-root v,Xy=Yy fred,blue} for general graph: 1.deal with irregularity by a metric in path coupling; 2.deal with cycles by self-avoiding walks; 3.deal with red/blue non-root vertices by a monotone argument;Xroot = red , Yroot = blue Δ-regular tree ∀ non-root v, Xv = Yv ∉ {red, blue} proposes a uniform random color σv∈[q]; update Xv to σv if for all v’s neighbors u: Xu≠σv ∧ σu≠Xv ∧ σu≠σv ; each v: for general graph: 1. deal with irregularity by a metric in path coupling; 2. deal with cycles by self-avoiding walks; 3. deal with red/blue non-root vertices by a monotone argument;