diff --git a/src/enzyme.jl b/src/enzyme.jl
index 74769504..ec674f61 100644
--- a/src/enzyme.jl
+++ b/src/enzyme.jl
@@ -110,7 +110,7 @@ function SparseEnzymeADJacobian(
   ncon,
   c!;
   x0::AbstractVector = rand(nvar),
-  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:direct}(),
+  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:direct}(postprocessing=true),
   detector::AbstractSparsityDetector = TracerSparsityDetector(),
   show_time::Bool = false,
   kwargs...,
@@ -130,7 +130,7 @@ function SparseEnzymeADJacobian(
   c!,
   J::SparseMatrixCSC{Bool, Int};
   x0::AbstractVector{T} = rand(nvar),
-  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:direct}(),
+  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:direct}(postprocessing=true),
   show_time::Bool = false,
   kwargs...,
 ) where {T}
@@ -187,7 +187,7 @@ function SparseEnzymeADHessian(
   ncon,
   c!;
   x0::AbstractVector = rand(nvar),
-  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:substitution}(),
+  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:substitution}(postprocessing=true),
   detector::AbstractSparsityDetector = TracerSparsityDetector(),
   show_time::Bool = false,
   kwargs...,
@@ -206,7 +206,7 @@ function SparseEnzymeADHessian(
   c!,
   H::SparseMatrixCSC{Bool, Int};
   x0::AbstractVector{T} = rand(nvar),
-  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:substitution}(),
+  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:substitution}(postprocessing=true),
   show_time::Bool = false,
   kwargs...,
 ) where {T}
diff --git a/src/sparse_hessian.jl b/src/sparse_hessian.jl
index f226ae0d..fa0be00d 100644
--- a/src/sparse_hessian.jl
+++ b/src/sparse_hessian.jl
@@ -22,7 +22,7 @@ function SparseADHessian(
   ncon,
   c!;
   x0::AbstractVector = rand(nvar),
-  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:direct}(),
+  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:direct}(postprocessing=true),
   detector::AbstractSparsityDetector = TracerSparsityDetector(),
   show_time::Bool = false,
   kwargs...,
@@ -41,7 +41,7 @@ function SparseADHessian(
   c!,
   H::SparseMatrixCSC{Bool, Int64};
   x0::S = rand(nvar),
-  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:direct}(),
+  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:direct}(postprocessing=true),
   show_time::Bool = false,
   kwargs...,
 ) where {S}
@@ -143,7 +143,7 @@ function SparseReverseADHessian(
   ncon,
   c!;
   x0::AbstractVector = rand(nvar),
-  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:substitution}(),
+  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:substitution}(postprocessing=true),
   detector::AbstractSparsityDetector = TracerSparsityDetector(),
   show_time::Bool = false,
   kwargs...,
@@ -162,7 +162,7 @@ function SparseReverseADHessian(
   c!,
   H::SparseMatrixCSC{Bool, Int};
   x0::AbstractVector{T} = rand(nvar),
-  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:substitution}(),
+  coloring_algorithm::AbstractColoringAlgorithm = GreedyColoringAlgorithm{:substitution}(postprocessing=true),
   show_time::Bool = false,
   kwargs...,
 ) where {T}
diff --git a/test/runtests.jl b/test/runtests.jl
index 44ccce21..89f97396 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -39,26 +39,54 @@ list_sparse_jac_backend =
 end
 
 list_sparse_hess_backend = (
-  (ADNLPModels.SparseADHessian, Dict(:coloring_algorithm => GreedyColoringAlgorithm{:direct}())),
+  (ADNLPModels.SparseADHessian,
+    "star coloring with postprocessing",
+    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:direct}(postprocessing=true)),
+  ),
+  (ADNLPModels.SparseADHessian,
+    "star coloring without postprocessing",
+    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:direct}(postprocessing=false)),
+  ),
+  (
+    ADNLPModels.SparseADHessian,
+    "acyclic coloring with postprocessing",
+    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:substitution}(postprocessing=true)),
+  ),
   (
     ADNLPModels.SparseADHessian,
-    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:substitution}()),
+    "acyclic coloring without postprocessing",
+    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:substitution}(postprocessing=false)),
+  ),
+  (
+    ADNLPModels.SparseReverseADHessian,
+    "star coloring with postprocessing",
+    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:direct}(postprocessing=true)),
+  ),
+  (
+    ADNLPModels.SparseReverseADHessian,
+    "star coloring without postprocessing",
+    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:direct}(postprocessing=false)),
   ),
   (
     ADNLPModels.SparseReverseADHessian,
-    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:direct}()),
+    "acyclic coloring with postprocessing",
+    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:substitution}(postprocessing=true)),
   ),
   (
     ADNLPModels.SparseReverseADHessian,
-    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:substitution}()),
+    "acyclic coloring without postprocessing",
+    Dict(:coloring_algorithm => GreedyColoringAlgorithm{:substitution}(postprocessing=false)),
+  ),
+  (ADNLPModels.ForwardDiffADHessian,
+    "default",
+    Dict(),
   ),
-  (ADNLPModels.ForwardDiffADHessian, Dict()),
 )
 
 @testset "Sparse Hessian" begin
-  for (backend, kw) in list_sparse_hess_backend
-    sparse_hessian(backend, kw)
-    sparse_hessian_nls(backend, kw)
+  for (backend, info, kw) in list_sparse_hess_backend
+    sparse_hessian(backend, info, kw)
+    sparse_hessian_nls(backend, info, kw)
   end
 end
 
diff --git a/test/sparse_hessian.jl b/test/sparse_hessian.jl
index ca130989..14c469e0 100644
--- a/test/sparse_hessian.jl
+++ b/test/sparse_hessian.jl
@@ -1,5 +1,5 @@
-function sparse_hessian(backend, kw)
-  @testset "Basic Hessian derivative with backend=$(backend) and T=$(T)" for T in (Float32, Float64)
+function sparse_hessian(backend, info, kw)
+  @testset "Basic Hessian derivative with backend=$(backend) -- $info -- T=$(T)" for T in (Float32, Float64)
     c!(cx, x) = begin
       cx[1] = x[1] - 1
       cx[2] = 10 * (x[2] - x[1]^2)
diff --git a/test/sparse_hessian_nls.jl b/test/sparse_hessian_nls.jl
index 2369fdd1..796e6567 100644
--- a/test/sparse_hessian_nls.jl
+++ b/test/sparse_hessian_nls.jl
@@ -1,5 +1,5 @@
-function sparse_hessian_nls(backend, kw)
-  @testset "Basic Hessian of residual derivative with backend=$(backend) and T=$(T)" for T in (
+function sparse_hessian_nls(backend, info, kw)
+  @testset "Basic Hessian of residual derivative with backend=$(backend) -- $info -- T=$(T)" for T in (
     Float32,
     Float64,
   )