gh-37181: Fix subtour elimination constraints in `longest_cycle`
    
An issue has been raised (see
#37028 (comment)) on
the formulation used to find the longest (induced) cycle. This was due
to the subtour elimination constraints that were not correct. We change
these constraints to fix this issue. The new constraints force to use
edges from the boundary of a subtour only when a vertex of that subtour
is selected.

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URL: #37181
Reported by: David Coudert
Reviewer(s): Travis Scrimshaw

Author: Release Manager

src/sage/graphs/generic_graph.py

@@ -7976,6 +7976,17 @@ def longest_cycle(self, induced=False, use_edge_labels=False,
 
         TESTS:
 
+        Check that the example from :issue:`37028` is fixed::
+
+            sage: d = {0: [4, 6, 10, 11], 1: [5, 7, 10, 11], 2: [8, 9, 10, 11],
+            ....:      3: [8, 9, 11], 4: [6, 10, 11], 5: [7, 10, 11],
+            ....:      6: [10, 11],  7: [10], 8: [10], 9: [11]}
+            sage: Z = Graph(d)
+            sage: Z.longest_cycle()
+            longest cycle: Graph on 9 vertices
+            sage: Z.longest_cycle(induced=True)
+            longest induced cycle: Graph on 5 vertices
+
         Small cases::
 
             sage: Graph().longest_cycle()
@@ -8085,8 +8096,8 @@ def F(e):
                                       constraint_generation=True)
 
         # We need one binary variable per vertex and per edge
-        vertex = p.new_variable(binary=True)
-        edge = p.new_variable(binary=True)
+        vertex = p.new_variable(binary=True, name='vertex')
+        edge = p.new_variable(binary=True, name='edge')
 
         # Objective function: maximize the size of the cycle
         p.set_objective(p.sum(weight(e) * edge[F(e)] for e in G.edge_iterator()))
@@ -8165,12 +8176,14 @@ def F(e):
 
                 # Add subtour elimination constraints
                 if directed:
-                    p.add_constraint(p.sum(edge[F(e)] for e in G.edge_boundary(c)), min=1)
                     c = set(c)
                     cbar = (v for v in G if v not in c)
-                    p.add_constraint(p.sum(edge[F(e)] for e in G.edge_boundary(cbar, c)), min=1)
+                    for u in c:
+                        p.add_constraint(vertex[u] <= p.sum(edge[F(e)] for e in G.edge_boundary(c)))
+                        p.add_constraint(vertex[u] <= p.sum(edge[F(e)] for e in G.edge_boundary(cbar, c)))
                 else:
-                    p.add_constraint(p.sum(edge[F(e)] for e in G.edge_boundary(c)), min=2)
+                    for u in c:
+                        p.add_constraint(2*vertex[u] <= p.sum(edge[F(e)] for e in G.edge_boundary(c)))
 
                 if induced:
                     # We eliminate this cycle