From dcb4316eca64acba81ed37ca84525c8a3382f303 Mon Sep 17 00:00:00 2001 From: didericis Date: Fri, 22 May 2026 12:09:13 -0400 Subject: [PATCH] Add Tutte-dual bridge-derivability test; rebuild artifacts experiments/test_tutte_bridge.py: bridge-derivability test for the dual of the 46-vertex Tutte graph (a 25-vertex non-intertwining triangulation, since the Tutte graph is non-Hamiltonian) -- the conjecture's first case beyond the n=21 Holton-McKay duals. Reuses the fast integer-state bridge engine: per source labelling with bipartite parity subgraphs, run a backward bridge-orbit BFS for an Even Level Graph witness. Co-Authored-By: Claude Opus 4.7 --- .../experiments/test_tutte_bridge.py | 171 ++++++++++++++++ .../paper.fdb_latexmk | 13 +- papers/even_level_graph_generators/paper.fls | 13 +- papers/even_level_graph_generators/paper.log | 190 ++++++------------ papers/even_level_graph_generators/paper.pdf | Bin 1093752 -> 1093752 bytes 5 files changed, 246 insertions(+), 141 deletions(-) create mode 100644 papers/even_level_graph_generators/experiments/test_tutte_bridge.py diff --git a/papers/even_level_graph_generators/experiments/test_tutte_bridge.py b/papers/even_level_graph_generators/experiments/test_tutte_bridge.py new file mode 100644 index 0000000..2841104 --- /dev/null +++ b/papers/even_level_graph_generators/experiments/test_tutte_bridge.py @@ -0,0 +1,171 @@ +"""Bridge-derivability test for the Tutte-graph dual. + +We take the classic 46-vertex Tutte graph (networkx `tutte_graph`), the +1946 counterexample to Tait's conjecture, dualize it to a 25-vertex +triangulation D, and ask whether D -- which is NOT an intertwining tree, +since its dual is non-Hamiltonian -- is a *bridge-derived level graph*. +This is the conjecture's first test outside the n=21 Holton-McKay duals. + +Method (reuses the fast integer-state bridge engine in fast_decide): +for each source s of D we form the BFS-parity labelling L (level mod 2), +keep only those L for which both parity subgraphs of D are bipartite +(a necessary condition, since bridge switches preserve bipartiteness), +and run a backward bridge-orbit BFS looking for an Even Level Graph whose +BFS-parity equals L. A witness proves D is bridge-derived; the BFS depth +of the witness is the number of bridge switches from the ELG to D (the +quantity tabulated for the n=21 duals). Not finding one within the +state/time budget is inconclusive, not a disproof. +""" +import sys +import os +import time +sys.path.insert(0, '/Users/didericis/Code/math-research/papers/' + 'level_resolutions_of_maximal_planar_graphs/experiments') +sys.path.insert(0, os.path.dirname(os.path.abspath(__file__))) +import networkx as nx +from tutte_dual_treecolor import dual_triangulation +from fast_bridge import EdgeCode +from fast_decide import expand_and_check + + +def characterize(D): + n = D.number_of_nodes() + m = D.number_of_edges() + is_tri = (m == 3 * n - 6) + planar, _ = nx.check_planarity(D) + # A triangulation is 4-connected iff it has no separating triangle, + # i.e. iff #triangles == #faces == 2n-4. Extra triangles are separating + # (dually: the cubic graph has a non-trivial 3-cut -> cyclically + # 3-connected, not 5-connected). + n_triangles = sum(nx.triangles(D).values()) // 3 + faces = 2 * n - 4 + sep_triangles = n_triangles - faces + return { + 'n': n, 'm': m, 'is_triangulation': is_tri, 'planar': planar, + 'triangles': n_triangles, 'faces': faces, + 'separating_triangles': sep_triangles, + 'four_connected': sep_triangles == 0, + } + + +def valid_parity_partitions_via_coloring(D, max_partitions=400, + max_colorings=2_000_000): + """A partition V = even u odd with both parity subgraphs bipartite is + exactly a proper 4-coloring of D grouped {0,1}|{2,3} (each parity class + is then 2-colored, hence bipartite). Enumerate proper 4-colorings by + backtracking (fixing colour of the first vertex to break the 4! colour + symmetry only partially) and collect the distinct induced partitions.""" + nodes = sorted(D.nodes()) + adj = {v: set(D.neighbors(v)) for v in nodes} + # colour high-degree vertices first -> heavier pruning + order = sorted(nodes, key=lambda v: -len(adj[v])) + colors = {} + parts = {} # canonical partition key -> labels dict + n_col = [0] + + def record(): + n_col[0] += 1 + even = frozenset(v for v in nodes if colors[v] in (0, 1)) + odd = frozenset(nodes) - even + if not odd: + return + key = min((even, odd), key=lambda s: (len(s), tuple(sorted(s)))) + if key not in parts: + parts[key] = {v: (0 if v in even else 1) for v in nodes} + + def bt(i): + if len(parts) >= max_partitions or n_col[0] >= max_colorings: + return + if i == len(order): + record() + return + v = order[i] + used = {colors[u] for u in adj[v] if u in colors} + cap = 1 if i == 0 else 4 # fix first vertex's colour to 0 + for c in range(cap): + if c in used: + continue + colors[v] = c + bt(i + 1) + del colors[v] + if len(parts) >= max_partitions or n_col[0] >= max_colorings: + return + + bt(0) + return list(parts.values()), n_col[0] + + +def search_partition(code, labels, n, cap, time_limit): + """Backward bridge-orbit BFS tracking depth. Returns + (status, n_states, depth_of_witness_or_None).""" + s0 = code.state0 + seen = {s0} + frontier = [s0] + depth = 0 + t0 = time.time() + while frontier and len(seen) < cap: + if time.time() - t0 > time_limit: + return 'timeout', len(seen), None + new = [] + for st in frontier: + wit, neigh = expand_and_check(st, code, labels, n) + if wit: + return 'found', len(seen), depth + for ns in neigh: + if ns not in seen: + seen.add(ns) + new.append(ns) + if len(seen) >= cap: + break + if len(seen) >= cap: + break + frontier = new + depth += 1 + return ('exhausted' if not frontier else 'capped'), len(seen), None + + +def main(cap=400_000, time_limit=120.0): + G = nx.tutte_graph() + D, _ = dual_triangulation(G) + info = characterize(D) + print('Tutte graph: |V|=%d (classic 46-vertex Tait counterexample)' + % G.number_of_nodes()) + print('Dual D: n=%(n)d, m=%(m)d, triangulation=%(is_triangulation)s, ' + 'planar=%(planar)s' % info) + print(' triangles=%(triangles)d, faces=%(faces)d, ' + 'separating triangles=%(separating_triangles)d, ' + '4-connected=%(four_connected)s' % info) + print(' [non-intertwining-tree by Thm: dual of a non-Hamiltonian C3CP]') + + code = EdgeCode(D.nodes()) + code.state0 = code.state_of(D) + n = info['n'] + + parts, n_col = valid_parity_partitions_via_coloring(D) + print('\n%d distinct valid parity partitions collected ' + '(from %d proper 4-colorings)' % (len(parts), n_col)) + if not parts: + print('No valid parity partition found: D is not bridge-derivable.') + return + + t0 = time.time() + for k, labels in enumerate(parts): + ev = sum(1 for v in labels if labels[v] == 0) + st, sz, depth = search_partition(code, labels, n, cap, time_limit) + tag = {'found': 'WITNESS', 'exhausted': 'exhausted (no witness)', + 'capped': 'CAP hit', 'timeout': 'TIMEOUT'}[st] + if st == 'found' or (k + 1) % 20 == 0 or st in ('capped', 'timeout'): + print(' partition %3d (even=%2d odd=%2d): %s [orbit>=%d, %.0fs]' + % (k, ev, n - ev, tag, sz, time.time() - t0)) + if st == 'found': + print('\n==> Tutte dual IS a bridge-derived level graph: ' + 'ELG witness at %d bridge switches (partition %d).' + % (depth, k)) + return + print('\n==> No ELG witness found over %d valid partitions within budget ' + '(cap=%d, %.0fs/partition). INCONCLUSIVE.' + % (len(parts), cap, time_limit)) + + +if __name__ == '__main__': + main() diff --git a/papers/even_level_graph_generators/paper.fdb_latexmk b/papers/even_level_graph_generators/paper.fdb_latexmk index f55db98..93cd83b 100644 --- a/papers/even_level_graph_generators/paper.fdb_latexmk +++ b/papers/even_level_graph_generators/paper.fdb_latexmk @@ -1,6 +1,6 @@ # Fdb version 3 -["pdflatex"] 1779463291 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779463293 - "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" 1779463291 18288 7a31bbe6e6a18516ee92fc4bff904881 "" +["pdflatex"] 1779464918 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779464920 + "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" 1779464918 21409 f57d3b5341eb97db9586774d14c6260d "" "/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df "" "/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" @@ -25,6 +25,7 @@ "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee "" + "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1248133631 35469 70d41d2b9ea31d5d813066df7c99281c "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 "" "/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 "" @@ -90,10 +91,10 @@ "fig_level_cycle.png" 1779389598 83736 ee54074ab1383a0dcc7fc20387e34bdc "" "fig_levels.png" 1779389598 88029 5564f46c0a183f3777727b651e7dc461 "" "fig_parity_subgraph.png" 1779389598 191771 f069aa94c8f49b3c7fd9c71426feff2d "" - "figures/n21_witnesses.png" 1779463145 347960 9cff26f0427b981c675f214ca5cc1513 "" - "paper.aux" 1779463292 7658 3054ab5e5e2c2d6239a5f60f25708981 "pdflatex" - "paper.out" 1779463292 1047 aeccb746cf11915bcb68f1e7ff8075a7 "pdflatex" - "paper.tex" 1779463291 18288 7a31bbe6e6a18516ee92fc4bff904881 "" + "figures/n21_duals.png" 1779463364 667947 fd52170c20399b0c2dff901831fad5d5 "" + "paper.aux" 1779464919 7535 db1f66072ebbe4d2fdf51a9fe4ad8e97 "pdflatex" + "paper.out" 1779464919 911 06539e3751da0b503943ca6640b71438 "pdflatex" + "paper.tex" 1779464918 21409 f57d3b5341eb97db9586774d14c6260d "" (generated) "paper.aux" "paper.log" diff --git a/papers/even_level_graph_generators/paper.fls b/papers/even_level_graph_generators/paper.fls index 46df0fc..3669937 100644 --- a/papers/even_level_graph_generators/paper.fls +++ b/papers/even_level_graph_generators/paper.fls @@ -545,6 +545,7 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7 INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm +INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map INPUT ./fig_levels.png INPUT ./fig_levels.png INPUT fig_levels.png @@ -555,7 +556,6 @@ INPUT ./fig_level_cycle.png INPUT fig_level_cycle.png INPUT ./fig_level_cycle.png INPUT ./fig_level_cycle.png -INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map INPUT ./fig_edge_switch.png INPUT ./fig_edge_switch.png INPUT fig_edge_switch.png @@ -566,11 +566,11 @@ INPUT ./fig_parity_subgraph.png INPUT fig_parity_subgraph.png INPUT ./fig_parity_subgraph.png INPUT ./fig_parity_subgraph.png -INPUT ./figures/n21_witnesses.png -INPUT ./figures/n21_witnesses.png -INPUT figures/n21_witnesses.png -INPUT ./figures/n21_witnesses.png -INPUT ./figures/n21_witnesses.png +INPUT ./figures/n21_duals.png +INPUT ./figures/n21_duals.png +INPUT figures/n21_duals.png +INPUT ./figures/n21_duals.png +INPUT ./figures/n21_duals.png INPUT paper.aux INPUT ./paper.out INPUT ./paper.out @@ -579,6 +579,7 @@ INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmcsc10. 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LaTeX Info: Redefining \pageref on input line 6101. - -(/usr/local/texlive/2022/texmf-dist/tex/latex/base/atbegshi-ltx.sty + (/usr/local/texlive/2022/texmf-dist/tex/latex/base/atbegshi-ltx.sty Package: atbegshi-ltx 2021/01/10 v1.0c Emulation of the original atbegshi package with kernel methods ) @@ -219,26 +190,20 @@ package with kernel methods \c@Hfootnote=\count279 ) Package hyperref Info: Driver (autodetected): hpdftex. - -(/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hpdftex.def + (/usr/local/texlive/2022/texmf-dist/tex/latex/hyperref/hpdftex.def File: hpdftex.def 2022-02-21 v7.00n Hyperref driver for pdfTeX - -(/usr/local/texlive/2022/texmf-dist/tex/latex/base/atveryend-ltx.sty -Package: atveryend-ltx 2020/08/19 v1.0a Emulation of the original atveryend pac -kage + (/usr/local/texlive/2022/texmf-dist/tex/latex/base/atveryend-ltx.sty +Package: atveryend-ltx 2020/08/19 v1.0a Emulation of the original atveryend package with kernel methods ) \Fld@listcount=\count280 \c@bookmark@seq@number=\count281 - -(/usr/local/texlive/2022/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty + (/usr/local/texlive/2022/texmf-dist/tex/latex/rerunfilecheck/rerunfilecheck.sty Package: rerunfilecheck 2019/12/05 v1.9 Rerun checks for auxiliary files (HO) - -(/usr/local/texlive/2022/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty + (/usr/local/texlive/2022/texmf-dist/tex/generic/uniquecounter/uniquecounter.sty Package: uniquecounter 2019/12/15 v1.4 Provide unlimited unique counter (HO) ) -Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 2 -86. +Package uniquecounter Info: New unique counter `rerunfilecheck' on input line 286. ) \Hy@SectionHShift=\skip57 ) (/usr/local/texlive/2022/texmf-dist/tex/latex/enumitem/enumitem.sty @@ -249,35 +214,28 @@ Package: enumitem 2019/06/20 v3.9 Customized lists \enit@inbox=\box53 \enit@count@id=\count282 \enitdp@description=\count283 -) -(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty +) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty Package: graphicx 2021/09/16 v1.2d Enhanced LaTeX Graphics (DPC,SPQR) - -(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty + (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty Package: graphics 2021/03/04 v1.4d Standard LaTeX Graphics (DPC,SPQR) - -(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty + (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty Package: trig 2021/08/11 v1.11 sin cos tan (DPC) -) -(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg +) (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg File: graphics.cfg 2016/06/04 v1.11 sample graphics configuration ) Package graphics Info: Driver file: pdftex.def on input line 107. - -(/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def + (/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def File: pdftex.def 2020/10/05 v1.2a Graphics/color driver for pdftex )) \Gin@req@height=\dimen155 \Gin@req@width=\dimen156 ) \c@theorem=\count284 - -(/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def + (/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX) \l__color_backend_stack_int=\count285 \l__pdf_internal_box=\box54 -) -(./paper.aux) +) (./paper.aux) \openout1 = `paper.aux'. 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