From b45c3d5510d6f98968879ba17c53835320dfb2a7 Mon Sep 17 00:00:00 2001 From: didericis Date: Fri, 22 May 2026 13:06:47 -0400 Subject: [PATCH] even_level: extend conjecture test to the cyclically-5-connected case (n=24) Add the n=24 result to the Even Level Graph Generators paper: the dual of the unique 44-vertex non-Hamiltonian cyclically-5-connected cubic planar graph (Holton-McKay Fig. 2.10) -- a 24-vertex 5-connected triangulation, the first conjecture test outside the 3-cut family -- is a bridge-derived level graph, two verified bridge switches from an Even Level Graph (source 19). - Generate the graph rather than transcribe it: plantri -c5 lists all 6833 5-connected 24-vertex triangulations; exactly one has a non-Hamiltonian dual, which also settles the uniqueness Holton-McKay left open at 44 vertices (cyclically-5-connected triangulation <=> dual cubic graph). - New abstract sentence + "cyclically-5-connected case: n=24" subsection, noting the classic 46-vertex Tutte graph is only cyclically 3-connected. - Figure 6 (figures/fig210_dual.png): the dual T, parity-coloured, with the two introduced bridge edges {6,19} and {20,22} in green (style of Fig. 5). - Experiments: test_fig210_dual_bridge.py (generate->filter->test pipeline), verify_fig210_witness.py (step-verifies the witness), draw_fig210_dual.py (figure), fig210_dual.g6 (the unique graph). paper.pdf rebuilt (10 pages). Co-Authored-By: Claude Opus 4.7 (1M context) --- .../experiments/draw_fig210_dual.py | 151 ++++++++++++++++++ .../experiments/fig210_dual.g6 | 1 + .../experiments/test_fig210_dual_bridge.py | 95 +++++++++++ .../experiments/verify_fig210_witness.py | 79 +++++++++ .../figures/fig210_dual.png | Bin 0 -> 152438 bytes papers/even_level_graph_generators/paper.aux | 40 ++--- .../paper.fdb_latexmk | 35 +++- papers/even_level_graph_generators/paper.fls | 148 ++++++++++++++++- papers/even_level_graph_generators/paper.log | 92 +++++++---- papers/even_level_graph_generators/paper.out | 3 +- papers/even_level_graph_generators/paper.pdf | Bin 1098280 -> 1234426 bytes papers/even_level_graph_generators/paper.tex | 72 ++++++++- 12 files changed, 657 insertions(+), 59 deletions(-) create mode 100644 papers/even_level_graph_generators/experiments/draw_fig210_dual.py create mode 100644 papers/even_level_graph_generators/experiments/fig210_dual.g6 create mode 100644 papers/even_level_graph_generators/experiments/test_fig210_dual_bridge.py create mode 100644 papers/even_level_graph_generators/experiments/verify_fig210_witness.py create mode 100644 papers/even_level_graph_generators/figures/fig210_dual.png diff --git a/papers/even_level_graph_generators/experiments/draw_fig210_dual.py b/papers/even_level_graph_generators/experiments/draw_fig210_dual.py new file mode 100644 index 0000000..ecec9c3 --- /dev/null +++ b/papers/even_level_graph_generators/experiments/draw_fig210_dual.py @@ -0,0 +1,151 @@ +"""Draw the dual of the unique 44-vertex non-Hamiltonian cyclically +5-connected cubic planar graph (Holton-McKay Fig. 2.10): a 24-vertex +5-connected triangulation T. Same style as draw_witnesses.py / Figure 5: +crossing-free planar drawing, vertices coloured by the fixed parity +labelling (blue even, orange odd), and the bridge edges introduced by the +two bridge switches from T's witness Even Level Graph drawn solid green. + +Writes ../figures/fig210_dual.png. +""" +import sys +import os +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 +import matplotlib +matplotlib.use('Agg') +import matplotlib.pyplot as plt +from matplotlib.lines import Line2D +from sage.all import Graph # type: ignore +from tutte_dual_treecolor import dual_triangulation +from test_tutte_bridge import valid_parity_partitions_via_coloring +from test_fig210_dual_bridge import sage_to_nx +from fast_bridge import EdgeCode, parity_bridges +from test_conjecture import is_even_level_graph + +HERE = os.path.dirname(os.path.abspath(__file__)) +FDIR = os.path.join(HERE, '..', 'figures') +EVEN_C = '#9ecae1' +ODD_C = '#fdae6b' + + +def build(): + g6 = open(os.path.join(HERE, 'fig210_dual.g6')).read().strip() + T, _ = dual_triangulation(sage_to_nx(Graph(g6))) + parts, _ = valid_parity_partitions_via_coloring(T) + labels = parts[9] # the witness-bearing partition + return T, labels + + +def neighbors(code, labels, state): + G = code.graph_of(state) + ok, emb = nx.check_planarity(G) + ea = {v: set() for v in code.nodes if labels[v] == 0} + oa = {v: set() for v in code.nodes if labels[v] == 1} + for u, v in G.edges(): + if labels[u] == labels[v]: + (ea if labels[u] == 0 else oa)[u].add(v) + (ea if labels[u] == 0 else oa)[v].add(u) + br = parity_bridges(ea) | parity_bridges(oa) + for u, v in G.edges(): + f1 = emb.traverse_face(u, v) + if len(f1) != 3: + continue + f2 = emb.traverse_face(v, u) + if len(f2) != 3: + continue + w = next(a for a in f1 if a not in (u, v)) + x = next(b for b in f2 if b not in (u, v)) + if w == x or G.has_edge(w, x) or labels[w] != labels[x]: + continue + if labels[u] == labels[v] and frozenset((u, v)) not in br: + continue + yield (state & ~(1 << code.bit(u, v))) | (1 << code.bit(w, x)) + + +def elg_src(code, labels, state): + G = code.graph_of(state) + for s in code.nodes: + cs = labels[s] + nb = set(G.neighbors(s)) + if not nb or any(labels[w] == cs for w in nb): + continue + ok, lv = is_even_level_graph(G, frozenset({s})) + if ok and all((lv[v] % 2 == 0) == (labels[v] == cs) for v in code.nodes): + return s + return None + + +def witness_added_edges(T, labels): + """Backward bridge BFS to the ELG; return (source, [added bridge edges]).""" + code = EdgeCode(T.nodes()) + s0 = code.state_of(T) + parent = {s0: None} + frontier = [s0] + W = None + while frontier and W is None: + nf = [] + for st in frontier: + if elg_src(code, labels, st) is not None: + W = st + break + for ns in neighbors(code, labels, st): + if ns not in parent: + parent[ns] = st + nf.append(ns) + if W: + break + frontier = nf + path = [] + c = W + while c is not None: + path.append(c) + c = parent[c] + # path[0]=ELG ... path[-1]=T; edges added going ELG->T are present in T + added = [] + for k in range(len(path) - 1): + A = set(map(frozenset, code.graph_of(path[k]).edges())) + B = set(map(frozenset, code.graph_of(path[k + 1]).edges())) + added.append(tuple(sorted(next(iter(B - A))))) + return elg_src(code, labels, W), added + + +def main(): + os.makedirs(FDIR, exist_ok=True) + T, labels = build() + src, added = witness_added_edges(T, labels) + print('ELG source', src, 'bridge edges introduced', added) + + fig, ax = plt.subplots(figsize=(7.5, 7.5)) + pos = nx.planar_layout(T) + colors = [EVEN_C if labels[v] == 0 else ODD_C for v in T.nodes()] + hl = {frozenset(e) for e in added} + plain = [e for e in T.edges() if frozenset(e) not in hl] + nx.draw_networkx_edges(T, pos, edgelist=plain, ax=ax, + edge_color='#b0b0b0', width=0.9) + nx.draw_networkx_edges(T, pos, edgelist=[tuple(e) for e in hl], ax=ax, + edge_color='#2ca02c', width=2.6) + nx.draw_networkx_nodes(T, pos, node_color=colors, node_size=240, + edgecolors='#444444', linewidths=0.6, ax=ax) + nx.draw_networkx_labels(T, pos, font_size=8, ax=ax) + ax.margins(0.12) + ax.axis('off') + handles = [ + Line2D([0], [0], marker='o', color='w', markerfacecolor=EVEN_C, + markeredgecolor='#444', markersize=9, label='even parity'), + Line2D([0], [0], marker='o', color='w', markerfacecolor=ODD_C, + markeredgecolor='#444', markersize=9, label='odd parity'), + Line2D([0], [0], color='#2ca02c', lw=2.6, label='bridge edge introduced'), + ] + fig.legend(handles=handles, loc='lower center', ncol=3, fontsize=9, + frameon=False) + fig.tight_layout(rect=(0, 0.05, 1, 1)) + out = os.path.join(FDIR, 'fig210_dual.png') + fig.savefig(out, dpi=160) + plt.close(fig) + print('wrote', out) + + +if __name__ == '__main__': + main() diff --git a/papers/even_level_graph_generators/experiments/fig210_dual.g6 b/papers/even_level_graph_generators/experiments/fig210_dual.g6 new file mode 100644 index 0000000..8be89ad --- /dev/null +++ b/papers/even_level_graph_generators/experiments/fig210_dual.g6 @@ -0,0 +1 @@ +ksP@@?PE?O?`@??_?O?A@?G??OG?O??G??A@??o??A???C@??E???@????O???E????G????OG???OG???G????B?????W????@?????A@????A@?????o?????G?????@@?????CC?????GG?????E??????@K diff --git a/papers/even_level_graph_generators/experiments/test_fig210_dual_bridge.py b/papers/even_level_graph_generators/experiments/test_fig210_dual_bridge.py new file mode 100644 index 0000000..b45ace1 --- /dev/null +++ b/papers/even_level_graph_generators/experiments/test_fig210_dual_bridge.py @@ -0,0 +1,95 @@ +"""Bridge-derivability test for the dual of Holton-McKay's Fig. 2.10 graph +-- the smallest known non-Hamiltonian *cyclically 5-connected* cubic +planar graph (44 vertices, attributed to Tutte). + +We obtain the graph by generation rather than transcription. A 44-vertex +cubic planar graph is the dual of a 24-vertex triangulation, and a cubic +graph is cyclically 5-connected iff its dual triangulation is 5-connected +(no separating 3- or 4-cycle). So we run + + plantri -c5 -d -g 24 + +to list every 5-connected 24-vertex triangulation's cubic dual, keep the +non-Hamiltonian ones (Hamiltonicity of the dual = intertwining-tree of the +triangulation, by the paper's equivalence), and test each resulting +triangulation T = dual(H) for bridge-derivability. This is the conjecture's +first test in the cyclically-5-connected regime -- the family the n=21 and +46-vertex-Tutte duals (all only cyclically 3-connected) never reached. + +Run after /tmp/nonham_duals.g6 has been produced by the Hamiltonicity +filter (lines: " "). +""" +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 sage.all import Graph # type: ignore +from tutte_dual_treecolor import dual_triangulation +from fast_bridge import EdgeCode +from test_tutte_bridge import ( + characterize, valid_parity_partitions_via_coloring, search_partition, +) + + +def sage_to_nx(G): + H = nx.Graph() + H.add_nodes_from(int(v) for v in G.vertices()) + H.add_edges_from((int(u), int(v)) for u, v in G.edges(labels=False)) + return H + + +def test_one(g6, cap=600_000, time_limit=180.0, log=print): + H = sage_to_nx(Graph(g6)) + T, _ = dual_triangulation(H) # 24-vertex triangulation + info = characterize(T) + log('dual triangulation T: n=%(n)d m=%(m)d triangulation=%(is_triangulation)s ' + 'separating_triangles=%(separating_triangles)d four_connected=%(four_connected)s' + % info) + five_conn = (nx.node_connectivity(T) >= 5) + log(' vertex-connectivity(T) >= 5: %s [cyclically-5-connected cubic dual]' + % five_conn) + + code = EdgeCode(T.nodes()) + code.state0 = code.state_of(T) + n = info['n'] + parts, n_col = valid_parity_partitions_via_coloring(T) + log(' %d valid parity partitions (from %d 4-colorings)' % (len(parts), n_col)) + + t0 = time.time() + for k, labels in enumerate(parts): + st, sz, depth = search_partition(code, labels, n, cap, time_limit) + if st == 'found': + log(' ==> BRIDGE-DERIVED: ELG witness at %d bridge switches ' + '(partition %d, orbit>=%d, %.0fs)' % (depth, k, sz, time.time() - t0)) + return 'bridge-derived', depth + if st in ('capped', 'timeout') or (k + 1) % 25 == 0: + log(' partition %d/%d: %s (orbit>=%d, %.0fs)' + % (k + 1, len(parts), st, sz, time.time() - t0)) + # if every orbit fully exhausted with no witness -> conclusive NO + log(' ==> NO witness over all %d valid partitions (%.0fs)' + % (len(parts), time.time() - t0)) + return 'no-witness', None + + +def main(): + path = '/tmp/nonham_duals.g6' + rows = [] + with open(path) as f: + for line in f: + line = line.strip() + if line: + idx, g6 = line.split(None, 1) + rows.append((int(idx), g6)) + print('%d non-Hamiltonian cyclically-5-connected 44-vertex duals to test\n' + % len(rows), flush=True) + for idx, g6 in rows: + print('### candidate (plantri index %d): %s' % (idx, g6), flush=True) + verdict, depth = test_one(g6) + print('### verdict: %s\n' % verdict, flush=True) + + +if __name__ == '__main__': + main() diff --git a/papers/even_level_graph_generators/experiments/verify_fig210_witness.py b/papers/even_level_graph_generators/experiments/verify_fig210_witness.py new file mode 100644 index 0000000..004639e --- /dev/null +++ b/papers/even_level_graph_generators/experiments/verify_fig210_witness.py @@ -0,0 +1,79 @@ +import sys +E='/Users/didericis/Code/math-research/papers/even_level_graph_generators/experiments' +L='/Users/didericis/Code/math-research/papers/level_resolutions_of_maximal_planar_graphs/experiments' +sys.path.insert(0,E); sys.path.insert(0,L) +import networkx as nx +from sage.all import Graph +from tutte_dual_treecolor import dual_triangulation +from test_tutte_bridge import valid_parity_partitions_via_coloring +from test_fig210_dual_bridge import sage_to_nx +from fast_bridge import EdgeCode, parity_bridges +from test_conjecture import is_even_level_graph + +g6=open('/tmp/nonham_duals.g6').read().split(None,1)[1].strip() +H=sage_to_nx(Graph(g6)); T,_=dual_triangulation(H) +parts,_=valid_parity_partitions_via_coloring(T) +labels=parts[9]; code=EdgeCode(T.nodes()); s0=code.state_of(T); n=24 + +def neighbors(state): + G=code.graph_of(state); ok,emb=nx.check_planarity(G) + ea={v:set() for v in code.nodes if labels[v]==0}; oa={v:set() for v in code.nodes if labels[v]==1} + for u,v in G.edges(): + if labels[u]==labels[v]: (ea if labels[u]==0 else oa)[u].add(v);(ea if labels[u]==0 else oa)[v].add(u) + br=parity_bridges(ea)|parity_bridges(oa); out=[] + for u,v in G.edges(): + f1=emb.traverse_face(u,v) + if len(f1)!=3: continue + f2=emb.traverse_face(v,u) + if len(f2)!=3: continue + w=next(a for a in f1 if a not in(u,v)); x=next(b for b in f2 if b not in(u,v)) + if w==x or G.has_edge(w,x) or labels[w]!=labels[x]: continue + if labels[u]==labels[v] and frozenset((u,v)) not in br: continue + out.append((state&~(1< T) +src=elg_src(W) +print('=== Fig 2.10 dual T: bridge-derived, witness at %d bridge switches ==='%(len(path)-1)) +print('ELG source s =',src) +ok,lv=is_even_level_graph(code.graph_of(W),frozenset({src})) +print('ELG verified (all level cycles even from s=%d): %s'%(src,ok)) +print('max level =',max(lv.values())) +allgood=True +for k in range(len(path)-1): + A=code.graph_of(path[k]); B=code.graph_of(path[k+1]) + EA=set(map(frozenset,A.edges())); EB=set(map(frozenset,B.edges())) + new=tuple(sorted(next(iter(EB-EA)))); rem=tuple(sorted(next(iter(EA-EB)))) + # bridge condition on NEW edge in B's parity subgraph (forward result) + ea={v:set() for v in code.nodes if labels[v]==0}; oa={v:set() for v in code.nodes if labels[v]==1} + for u,v in B.edges(): + if labels[u]==labels[v]: (ea if labels[u]==0 else oa)[u].add(v);(ea if labels[u]==0 else oa)[v].add(u) + br=parity_bridges(ea)|parity_bridges(oa) + if labels[new[0]]==labels[new[1]]: + valid=frozenset(new) in br; kind='same-parity bridge in %s subgraph'%('even' if labels[new[0]]==0 else 'odd') + else: + valid=True; kind='cross-parity (enters neither subgraph)' + allgood &= valid + print(' switch %d: remove level-edge %s, add %s [%s] valid=%s'%(k+1,rem,new,kind,valid)) +print('ALL STEPS VALID BRIDGE SWITCHES:',allgood) +print('T is intertwining 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\newlabel{def:edge-switch}{{2.4}{2}{Edge switch}{theorem.2.4}{}} -\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Outerplanarity of level components}}{2}{section.3}\protected@file@percent } -\newlabel{sec:outerplanar-components}{{3}{2}{Outerplanarity of level components}{section.3}{}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level.}}{3}{figure.1}\protected@file@percent } \newlabel{fig:levels}{{1}{3}{BFS levels from the degree-$3$ vertex source $S = \{4\}$. The source is level $0$, its three neighbours are level $1$, and the remaining vertices are level $2$. Colour encodes the level}{figure.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces A level cycle in the triangulation of Figure\nonbreakingspace \ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$.}}{3}{figure.2}\protected@file@percent } \newlabel{fig:level-cycle}{{2}{3}{A level cycle in the triangulation of Figure~\ref {fig:levels}. The triangle $1\!-\!2\!-\!3$ is a simple cycle whose three vertices all lie at level $1$, so it is a level cycle at level $1$}{figure.2}{}} +\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{Outerplanarity of level components}}{3}{section.3}\protected@file@percent } +\newlabel{sec:outerplanar-components}{{3}{3}{Outerplanarity of level components}{section.3}{}} \newlabel{thm:outerplanar-component}{{3.1}{3}{}{theorem.3.1}{}} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces An edge switch on the level cycle of Figure\nonbreakingspace \ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$.}}{4}{figure.3}\protected@file@percent } \newlabel{fig:edge-switch}{{3}{4}{An edge switch on the level cycle of Figure~\ref {fig:level-cycle}. The chosen cycle edge $1\!-\!2$ is shared by the triangular faces $(0,1,2)$ and $(1,2,4)$; the switch deletes $1\!-\!2$ (red, left) and inserts $0\!-\!4$ (green, right). Vertex colours indicate the original levels in $G$}{figure.3}{}} \@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Parity subgraphs of $G' = T$ with respect to the level structure of Figure\nonbreakingspace \ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \nonscript \mskip -\medmuskip \mkern 5mu\mathbin {\mathgroup \symoperators mod}\penalty 900 \mkern 5mu\nonscript \mskip -\medmuskip 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. Right: the odd parity subgraph $O_{G,S}(G')$, induced on $\{1, 2, 3\}$; the highlighted triangle shows that $O_{G,S}(G')$ is not bipartite for this choice of $G'$.}}{4}{figure.4}\protected@file@percent } \newlabel{fig:parity-subgraph}{{4}{4}{Parity subgraphs of $G' = T$ with respect to the level structure of Figure~\ref {fig:levels} (here we take $G = G' = T$). Left: $T$ with vertices coloured by $\ell _G \bmod 2$ (blue $=$ even, orange $=$ odd). Middle: the even parity subgraph $E_{G,S}(G')$, induced on $\{0, 4, 5, 6\}$; only edges with both endpoints even appear. 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The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\DOTSB \sum@ \slimits@ _G \genfrac {}{}{}1{4E}{|\mathrm {Aut}(G)|}\,s(G)$.}}{5}{table.1}\protected@file@percent } -\newlabel{tab:elg-counts}{{1}{5}{Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\sum _G \tfrac {4E}{|\mathrm {Aut}(G)|}\,s(G)$}{table.1}{}} \newlabel{def:derived-level-graph}{{4.3}{5}{Derived level graph}{theorem.4.3}{}} -\citation{holton-mckay} -\newlabel{def:bridge-switch}{{4.4}{6}{Bridge switch}{theorem.4.4}{}} +\newlabel{def:bridge-switch}{{4.4}{5}{Bridge switch}{theorem.4.4}{}} +\@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\DOTSB \sum@ \slimits@ _G \genfrac {}{}{}1{4E}{|\mathrm {Aut}(G)|}\,s(G)$.}}{6}{table.1}\protected@file@percent } +\newlabel{tab:elg-counts}{{1}{6}{Even Level Graph counts for $4 \leq n \leq 11$. The \emph {triangulations} column is the number of plane-triangulation iso classes (OEIS A000109). \emph {ELG iso classes} counts pairs $(G, S)$ up to isomorphism; \emph {flag-rooted ELGs} is the automorphism-free count $\sum _G \tfrac {4E}{|\mathrm {Aut}(G)|}\,s(G)$}{table.1}{}} \newlabel{def:bridge-derived-level-graph}{{4.5}{6}{Bridge-derived level graph}{theorem.4.5}{}} \newlabel{def:intertwining-tree}{{4.6}{6}{Intertwining tree}{theorem.4.6}{}} \newlabel{thm:intertwining-iff-hamiltonian-dual}{{4.7}{6}{}{theorem.4.7}{}} -\newlabel{conj:every-triangulation-derived}{{4.8}{6}{}{theorem.4.8}{}} +\citation{holton-mckay} +\citation{holton-mckay} +\newlabel{conj:every-triangulation-derived}{{4.8}{7}{}{theorem.4.8}{}} +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The cyclically-$5$-connected case: $n = 24$}}{7}{section*.3}\protected@file@percent } +\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified.}}{8}{table.2}\protected@file@percent } +\newlabel{tab:n21}{{2}{8}{The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified}{table.2}{}} +\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{8}{figure.5}\protected@file@percent } +\newlabel{fig:n21-duals}{{5}{8}{The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge}{figure.5}{}} \bibcite{holton-mckay}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{14.69437pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} -\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{The boundary case $n = 21$}}{7}{section*.2}\protected@file@percent } -\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{7}{section*.3}\protected@file@percent } -\@writefile{lot}{\contentsline {table}{\numberline {2}{\ignorespaces The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified.}}{8}{table.2}\protected@file@percent } -\newlabel{tab:n21}{{2}{8}{The six Holton--McKay duals at $n = 21$, the first triangulations that are not intertwining trees. Each is a bridge-derived level graph: duals $1$ and $2$ are Even Level Graphs outright (zero switches), and the remaining four reach an Even Level Graph in $1$--$4$ bridge switches. All witnesses are step-verified}{table.2}{}} -\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge.}}{8}{figure.5}\protected@file@percent } -\newlabel{fig:n21-duals}{{5}{8}{The six Holton--McKay duals, drawn as crossing-free planar graphs and coloured by parity (blue even, orange odd, with respect to the fixed level-parity labelling). The solid green edges are the bridge edges introduced by the bridge switches from each dual's witness Even Level Graph. Each green edge is a bridge of its parity subgraph, so no new cycle -- and in particular no odd cycle -- is created; duals $1$ and $2$ coincide with their Even Level Graphs and have no added edge}{figure.5}{}} -\gdef \@abspage@last{8} +\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{section*.4}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.\nonbreakingspace 2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created.}}{10}{figure.6}\protected@file@percent } +\newlabel{fig:n24-dual}{{6}{10}{The $24$-vertex dual $T$ of the unique $44$-vertex non-Hamiltonian cyclically $5$-connected cubic planar graph (Holton--McKay Fig.~2.10), drawn crossing-free and coloured by the fixed parity labelling (blue even, orange odd). $T$ is $5$-connected and not an intertwining tree, yet is a bridge-derived level graph: the two solid green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by the two bridge switches carrying its witness Even Level Graph (source $19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, 19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and in particular no odd cycle, is created}{figure.6}{}} +\gdef \@abspage@last{10} diff --git a/papers/even_level_graph_generators/paper.fdb_latexmk b/papers/even_level_graph_generators/paper.fdb_latexmk index e603040..2b74724 100644 --- a/papers/even_level_graph_generators/paper.fdb_latexmk +++ b/papers/even_level_graph_generators/paper.fdb_latexmk @@ -1,6 +1,25 @@ # Fdb version 3 -["pdflatex"] 1779466997 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779466999 - "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" 1779466995 23442 551d7ef123af98c3c5f0ca2b7125c5d5 "" +["pdflatex"] 1779469001 "/Users/didericis/Code/math-research/papers/even_level_graph_generators/paper.tex" "paper.pdf" "paper" 1779469003 + 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so every triangulation on at most $20$ vertices is an intertwining tree and the first possible failures occur at $n = 21$, at the six duals of the Holton--McKay graphs. We verify that all six are bridge-derived level graphs, confirming the conjecture in its first -nontrivial case. +nontrivial case. Pushing further, we identify by exhaustive generation +the unique $44$-vertex non-Hamiltonian \emph{cyclically $5$-connected} +cubic planar graph -- settling a uniqueness question Holton--McKay left +open -- whose $24$-vertex $5$-connected dual is the first test of the +conjecture outside the $3$-cut family; it too is a bridge-derived level +graph, two bridge switches from an Even Level Graph. \end{abstract} \maketitle @@ -528,6 +533,71 @@ with their Even Level Graphs and have no added edge.} \label{fig:n21-duals} \end{figure} +\subsection*{The cyclically-$5$-connected case: $n = 24$} + +The six $n = 21$ duals all carry non-trivial $3$-cuts in the cubic +picture; dually, each contains a separating triangle, so each is built +from smaller pieces and lies in the most reducible part of the +non-Hamiltonian world. (The famous $46$-vertex Tutte graph is no +improvement here: it too is only cyclically $3$-connected, and its +$25$-vertex dual has separating triangles.) The genuinely new regime is +the \emph{cyclically $5$-connected} one, dual to a $5$-connected +triangulation -- no separating $3$- or $4$-cycle, hence nothing to +decompose along. By Holton--McKay, the smallest non-Hamiltonian +cyclically $5$-connected cubic planar graph has $44$ vertices (Fig.~2.10 +of~\cite{holton-mckay}, attributed to Tutte; minimality due to +Faulkner--Younger), and its dual is a $24$-vertex $5$-connected +triangulation. + +We obtain this graph by generation rather than transcription. A +$44$-vertex cubic planar graph is the dual of a $24$-vertex +triangulation, and a cubic graph is cyclically $5$-connected if and only +if its dual triangulation is $5$-connected. Enumerating all $5$-connected +triangulations on $24$ vertices (\texttt{plantri -c5}, $6833$ of them) +and testing each dual for Hamiltonicity, we find that \emph{exactly one} +has a non-Hamiltonian dual. This both produces the graph and, granting +the correctness of the generator and the Hamiltonicity test, settles the +uniqueness question Holton--McKay left open: there is a unique +non-Hamiltonian cyclically $5$-connected cubic planar graph on $44$ +vertices. + +Let $T$ be its dual: a $24$-vertex triangulation with vertex connectivity +$5$ and no separating triangle, and -- since its dual is non-Hamiltonian +-- not an intertwining tree. We find that $T$ is nonetheless a +bridge-derived level graph. Of its $333$ valid parity partitions most are +useless: their backward bridge-orbits exceed $8 \times 10^5$ states with +no Even Level Graph in sight. But one partition has a backward orbit of +only $4678$ states containing an Even Level Graph (source $s = 19$, +maximum level $4$) at depth $2$. The two bridge switches carrying that +Even Level Graph to $T$ are +\[ + \text{remove } \{16,21\},\ \text{add } \{20,22\} + \quad\text{and}\quad + \text{remove } \{15,18\},\ \text{add } \{6,19\}, +\] +each adding a same-parity edge that is a bridge of the (odd, resp.\ even) +parity subgraph; both steps have been verified to be valid bridge +switches. So the disjunction holds for $T$ through the bridge-derived +disjunct, and the ``one good partition suffices'' phenomenon seen at +$n = 21$ persists into the cyclically $5$-connected regime -- the first +test of the conjecture genuinely outside the $3$-cut family. + +\begin{figure}[ht] +\centering +\includegraphics[width=0.7\textwidth]{figures/fig210_dual.png} +\caption{The $24$-vertex dual $T$ of the unique $44$-vertex +non-Hamiltonian cyclically $5$-connected cubic planar graph +(Holton--McKay Fig.~2.10), drawn crossing-free and coloured by the fixed +parity labelling (blue even, orange odd). $T$ is $5$-connected and not an +intertwining tree, yet is a bridge-derived level graph: the two solid +green edges $\{6,19\}$ and $\{20,22\}$ are the bridge edges introduced by +the two bridge switches carrying its witness Even Level Graph (source +$19$) to $T$. Each green edge is a bridge of its parity subgraph -- $\{6, +19\}$ in the even subgraph, $\{20,22\}$ in the odd -- so no new cycle, and +in particular no odd cycle, is created.} +\label{fig:n24-dual} +\end{figure} + \begin{thebibliography}{9} \bibitem{holton-mckay}