experiments/search_smaller_counterexample.py enumerates 3-connected
cubic planar graphs via graphs.planar_graphs(n, min_deg=3, min_conn=2)
(filtering to cubic), then for each graph tries every proper
3-edge-colouring (backtracking with symmetry-break on first edge),
computes h_φ via the CW rotation from sage's planar embedding, and
checks whether some pair of intersecting Kempe cycles K_{a,b} and
K_{a,c} are both constant-Heawood.
Results (up to n=10 in initial run):
n= 4: K_4 itself. Coloring (1,2)=red, (3,4)=red, (1,3)=blue,
(2,4)=blue, (1,4)=green, (2,3)=green; sage's CW embedding
gives h_φ ≡ -1 on all 4 vertices. K_{red,blue} = 4-cycle
1-2-4-3 and K_{red,green} = 4-cycle 1-2-3-4 share both red
edges; both constant.
n= 6: no counterexample (only the triangular prism).
n= 8: a 12-edge cubic planar graph (graph6 G}GOW[) on 8 vertices.
Both Kempe cycles are 8-cycles visiting every vertex.
n=10: 8 cubic planar graphs checked, no counterexample.
So K_4 is the smallest counterexample to Conjecture 5.5 as stated,
but both K_4 and the n=8 example are structurally trivial: K_0 and
K_1 jointly cover V(H). The user's 40-vertex counterexample (paper
Figure) is the smallest non-trivial example found so far, with 24
vertices outside V(K_0) ∪ V(K_1).
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
didericis
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