dual_decomposition: Conj 3.6 (face/Kempe witness) and constructive lift
Paper: - Lemmas 3.4 (exactly one match) and 3.5 (all-distinct exists for 4-colourable G) replace the earlier conjecture; both have proofs. - Add Conjecture 3.6: every proper 3-edge-colouring of a counterexample's reduced dual has a face with two same-colour edges that share a Kempe cycle with the merged edge, neither of them being the merged edge. Experiments (all under experiments/): - search_conj_3_6_counterexample.py: finds n=14 tri#1 i_red=0 where the algorithm's phi_t* sits in a Kempe class with no all-distinct colouring (disproves an earlier formulation). - check_kempe_class.py / check_kempe_class_invariance.py / check_kempe_class_monotone.py: Kempe-class counts on H_1 and H_t* for small triangulations; neither monotonicity direction holds. - check_all_distinct_exists.py: even in the conj-3.6 disproof case, H_t* itself admits all-distinct colourings in the *other* Kempe class. - check_constrained_feasibility.py: literal H_t*-interpretation of C1 + K0 + K1 is empirically unsatisfiable (gap in proof strategy noted). - check_conj_face_kempe.py / check_conj_face_kempe_n15.py: test Conj 3.6 on chord-apex+Kempe colourings of reduced duals at n=12, 14, 15; 216/216 colourings on n=14 satisfy the conjecture, others vacuous. - draw_step1_conj36.py: figure showing a Conj 3.6 witness on H_1 with two new vertices on the witness edges and a new red bridge between them. - draw_step1_conj36_recolored.py: same but with the Kempe cycle recoloured alternately from merged so propriety holds. - draw_lift_to_Gprime.py: lifts the modified+recoloured H_1 back to a proper 3-edge-colouring of the modified G' (24+2 vertices, 39 edges, same Tutte layout as figure 3's first graphic so positions line up). Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
|
||||
Output written on paper.pdf (9 pages, 977618 bytes).
|
||||
PDF statistics:
|
||||
137 PDF objects out of 1000 (max. 8388607)
|
||||
72 compressed objects within 1 object stream
|
||||
140 PDF objects out of 1000 (max. 8388607)
|
||||
74 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
51 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Reference in New Issue
Block a user