△-regular tree each v: proposes a uniform random color o,E[g]; update Xy to oy if for all v's neighbors u: X≠GvAO计XAG≠0v; Xroot red,Yroot blue V non-root v,Xy=Yy fred,blue} coupling:coupling the proposals (so that ( root Pr≠l≤1-(-)(-) nonroot le≠Hsg-8)()》 ≠%a+Pmx≤1-（-台)(-)+ga(-) -1 non-root u (assume q≥a△) s1-e加(-d-a2)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: coupling: coupling the proposals (σX , σY) so that (X, Y ) (￾ X ,￾ Y ) ￾! (X0 , Y 0 ) Pr[X0 root 6= Y 0 root]  1 ￾ ✓ 1 ￾ ￾ q ◆ ✓1 ￾ 2 q ◆￾ non-root u at level l: Pr[X0 root 6= Y 0 root] + X non-root u Pr[X0 u 6= Y 0 u]  1 ￾ ✓ 1 ￾ ￾ q ◆ ✓1 ￾ 2 q ◆￾ + ￾ q ￾ 2￾ ✓ 1 ￾ 2 q ◆￾￾1 Pr[X0 u 6= Y 0 u]  1 q ✓ 1 ￾ 2 q ◆￾￾1 ✓2 q ◆`￾1 root:  1 ￾ e￾2/↵ ✓ 1 ￾ 1 ↵ ￾ 1 ↵ ￾ 2 ◆ (assume ) q ￾ ↵￾