From 20f19f086937c369b83935f5af77f1abb918294f Mon Sep 17 00:00:00 2001 From: didericis Date: Fri, 22 May 2026 20:13:24 -0400 Subject: [PATCH] even_level: extend to n=25 -- second internally-6-connected core, also bridge-derived Enumerate non-Hamiltonian cyclically-5-connected cubic planar graphs by running plantri -c5 -d for n in {23,25,26} (n=24 already in the previous commit) and filtering for non-Hamiltonian dual: n=23 -> 0 of 1970 (recomputes Faulkner-Younger minimality) n=24 -> 1 of 6833 (the Tutte/Fig 2.10 graph) n=25 -> 1 of 23384 (new; unique 46-vertex one) n=26 -> 0 of 82625 Both T (n=24) and T_25 (n=25) verified internally 6-connected by exhaustive 5-cut scan: every 5-cut is the neighborhood of a degree-5 vertex. This is the strongest connectivity a planar triangulation can have and the level at which Birkhoff-style reductions terminate, so both are genuinely irreducible bases of any decomposition argument. T_25 is also bridge-derived: witness Even Level Graph from source 24 (max level 4) at depth 2, orbit only 3114 states. Forward switches: remove {21,23} add {22,24}; remove {3,5} add {1,6}. Both adds are bridges of the even parity subgraph. Same witness signature as T (minimum total Betti, tiny orbit, depth 2). New subsection "Beyond n=24: enumeration and the next 5-connected core", abstract extended, new Figure 7 (core_n25_dual.png). Reproducibility scripts: draw_core_witness.py and verify_core_witness.py (both parametrized so they work on any 5-conn non-Ham-dual core's g6). Co-Authored-By: Claude Opus 4.7 (1M context) --- .../experiments/core_n25_dual.g6 | 1 + .../experiments/draw_core_witness.py | 171 ++++++++++++++++++ .../experiments/verify_core_witness.py | 101 +++++++++++ .../figures/core_n25_dual.png | Bin 0 -> 167150 bytes papers/even_level_graph_generators/paper.aux | 29 +-- papers/even_level_graph_generators/paper.log | 120 ++++++++---- papers/even_level_graph_generators/paper.out | 5 +- papers/even_level_graph_generators/paper.pdf | Bin 1245166 -> 1384388 bytes papers/even_level_graph_generators/paper.tex | 88 ++++++++- 9 files changed, 457 insertions(+), 58 deletions(-) create mode 100644 papers/even_level_graph_generators/experiments/core_n25_dual.g6 create mode 100644 papers/even_level_graph_generators/experiments/draw_core_witness.py create mode 100644 papers/even_level_graph_generators/experiments/verify_core_witness.py create mode 100644 papers/even_level_graph_generators/figures/core_n25_dual.png diff --git a/papers/even_level_graph_generators/experiments/core_n25_dual.g6 b/papers/even_level_graph_generators/experiments/core_n25_dual.g6 new file mode 100644 index 0000000..5ddb2a3 --- /dev/null +++ b/papers/even_level_graph_generators/experiments/core_n25_dual.g6 @@ -0,0 +1 @@ +msP@@?PE?O?`@??_?O?A@?G??OG?O??G??A@??o??A???C@??C???A????_???C????o????_????_G???OC???E?????_????A?_???K?????K?????C?????@_G????B??????CC?????G??????GG?????CC?????@@??????BG diff --git a/papers/even_level_graph_generators/experiments/draw_core_witness.py b/papers/even_level_graph_generators/experiments/draw_core_witness.py new file mode 100644 index 0000000..4190d7a --- /dev/null +++ b/papers/even_level_graph_generators/experiments/draw_core_witness.py @@ -0,0 +1,171 @@ +"""Draw a 5-connected non-Hamiltonian-dual core (= the dual of a non-Hamiltonian +cyclically 5-connected cubic planar graph) as a parity-coloured planar graph +with the bridge edges introduced by its witness Even Level Graph highlighted. +Style matches Fig. 6 (the n=24 core). + +The script picks the first valid parity partition of minimal total Betti for +which a backward bridge-orbit search returns an Even Level Graph, and prints +the witness data (source, added bridge edges). + +Usage: + sage -python draw_core_witness.py +""" +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 + +EVEN_C = '#9ecae1' +ODD_C = '#fdae6b' + + +def betti(T, labels): + A = [v for v in T if labels[v] == 0] + B = [v for v in T if labels[v] == 1] + GA, GB = T.subgraph(A), T.subgraph(B) + return (GA.number_of_edges() - len(A) + nx.number_connected_components(GA)) + \ + (GB.number_of_edges() - len(B) + nx.number_connected_components(GB)) + + +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 find_witness(T, labels, cap): + code = EdgeCode(T.nodes()) + s0 = code.state_of(T) + parent = {s0: None} + frontier = [s0] + W = None + while frontier and W is None and len(parent) < cap: + 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 len(parent) >= cap: + break + if W or len(parent) >= cap: + break + if W: + break + frontier = nf + if W is None: + return None + path = [] + c = W + while c is not None: + path.append(c) + c = parent[c] + 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, len(path) - 1 + + +def main(): + if len(sys.argv) < 3: + print('usage: draw_core_witness.py ') + sys.exit(1) + in_g6_path, out_png = sys.argv[1], sys.argv[2] + g6 = open(in_g6_path).read().strip() + T, _ = dual_triangulation(sage_to_nx(Graph(g6))) + parts, _ = valid_parity_partitions_via_coloring(T) + order = sorted(range(len(parts)), key=lambda k: betti(T, parts[k])) + chosen = None + for k in order[:200]: + r = find_witness(T, parts[k], cap=40000) + if r is not None: + chosen = (k, parts[k], r) + break + if chosen is None: + print('no witness found in scanned partitions; aborting') + sys.exit(2) + k, labels, (src, added, depth) = chosen + print('partition=%d source=%d depth=%d added=%s' % (k, src, depth, added)) + + fig, ax = plt.subplots(figsize=(8, 8)) + 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)) + os.makedirs(os.path.dirname(out_png), exist_ok=True) + fig.savefig(out_png, dpi=160) + plt.close(fig) + print('wrote', out_png) + + +if __name__ == '__main__': + main() diff --git a/papers/even_level_graph_generators/experiments/verify_core_witness.py b/papers/even_level_graph_generators/experiments/verify_core_witness.py new file mode 100644 index 0000000..651ed89 --- /dev/null +++ b/papers/even_level_graph_generators/experiments/verify_core_witness.py @@ -0,0 +1,101 @@ +"""Step-verify the bridge-derivability witness for a 5-connected +non-Hamiltonian-dual core. Same outputs as verify_fig210_witness.py +(parity partition, ELG source/max level, both removed and added edges per +forward switch, bridge-condition validity check) but parametrized so it +works on any core's graph6. + +Usage: + sage -python verify_core_witness.py +""" +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 +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 +from draw_core_witness import betti, neighbors, elg_src, find_witness + + +def main(): + if len(sys.argv) < 2: + print('usage: verify_core_witness.py ') + sys.exit(1) + g6 = open(sys.argv[1]).read().strip() + T, _ = dual_triangulation(sage_to_nx(Graph(g6))) + parts, _ = valid_parity_partitions_via_coloring(T) + order = sorted(range(len(parts)), key=lambda k: betti(T, parts[k])) + + chosen = None + for k in order[:200]: + r = find_witness(T, parts[k], cap=40000) + if r is not None: + chosen = (k, parts[k], r) + break + assert chosen is not None, 'no witness found' + k, labels, (src, _, _) = chosen + + 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] + + ok, lv = is_even_level_graph(code.graph_of(W), frozenset({src})) + print('partition %d, source %d, max level %d, depth %d, ELG verified %s' + % (k, src, max(lv.values()), len(path) - 1, ok)) + print('parity even =', sorted(v for v in T if labels[v] == 0)) + print('parity odd =', sorted(v for v in T if labels[v] == 1)) + + all_valid = True + for i in range(len(path) - 1): + A = set(map(frozenset, code.graph_of(path[i]).edges())) + B = set(map(frozenset, code.graph_of(path[i + 1]).edges())) + new = tuple(sorted(next(iter(B - A)))) + rem = tuple(sorted(next(iter(A - B)))) + Gp = code.graph_of(path[i + 1]) + 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 Gp.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 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z!k*d%((57%SO*pVCM7mOP)(5_GS4Gr&cm&sM6+>`#=kAF%`rg#`#Us)Y;=4DW5xL6 zanKsHqI#wNJM~g!+F=j|LL*e+Z80Vm=~zG)zMTg2d)JM67m@|9zMjn1l$lA(mtNn; z2P>ph2!CLb$p#lk?0lJ*n!Q}5Y-$DigyNL8?9b9IYTXtJVX$vPz5lR`*aT)XMp#15 zG!}A~_d3fKX(DQp;5ZJ{#qt7HF$YqEGd7Pl#QrVcR2YWq~oYc7}1s|c36YaA3!OEuCHC{l0OM5oDduh}uN4R;0?Ysecm zTb3&&NjkAw&_1koR-`G`VMDa2%Hs$WQ!+*kNa5>Lp6Hn?BgKaqzV39DS zoZ+*an&c*OG7W`7cgcfr4a8zz1)3C4*kz5HLZOJ5VoDk#+mvWj$B~_Z`j8iAqN&ttx~1f27Pbs{jB1 diff --git a/papers/even_level_graph_generators/paper.tex b/papers/even_level_graph_generators/paper.tex index c26e9e6..1af0c65 100644 --- a/papers/even_level_graph_generators/paper.tex +++ b/papers/even_level_graph_generators/paper.tex @@ -112,7 +112,15 @@ 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. +graph, two bridge switches from an Even Level Graph. Iterating the same +generation procedure up to $n = 26$ produces, at $n = 25$, a second +non-Hamiltonian cyclically-$5$-connected cubic planar graph (likewise +unique at its size); its dual is verified \emph{internally +$6$-connected} -- the strongest connectivity possible for a planar +triangulation, and the level at which Birkhoff-style reductions +terminate -- and likewise bridge-derived, again at depth $2$. Both known +internally-$6$-connected non-Hamiltonian-dual cores thus satisfy the +conjecture, with identical witness signature. \end{abstract} \maketitle @@ -586,8 +594,8 @@ 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 +$5$ (in fact internally $6$-connected, verified in the next subsection), +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 @@ -622,6 +630,80 @@ in particular no odd cycle, is created.} \label{fig:n24-dual} \end{figure} +\subsection*{Beyond $n = 24$: enumeration and the next $5$-connected core} + +The graph $T$ of the previous subsection is one core; iterating the same +generation procedure produces the rest. Below, the third column counts +$5$-connected triangulations on $n$ vertices (\texttt{plantri -c5} $n$), +and the fourth filters them by Hamiltonicity of the cubic dual: +\begin{center} +\begin{tabular}{cccc} +$n$ & dual $|V|$ & $5$-connected triangulations & non-Hamiltonian dual \\\hline +$23$ & $42$ & $1{,}970$ & $0$ \\ +$24$ & $44$ & $6{,}833$ & $1$ \quad (Holton--McKay Fig.~2.10) \\ +$25$ & $46$ & $23{,}384$ & $1$ \\ +$26$ & $48$ & $82{,}625$ & $0$ \\ +\end{tabular} +\end{center} +The $n = 23$ row recomputes Faulkner--Younger's minimality (no cyclically +$5$-connected non-Hamiltonian cubic planar graph below $44$ vertices). At +$n = 25$ we find a single new cyclically $5$-connected non-Hamiltonian +cubic planar graph on $46$ vertices; its dual we call $T_{25}$, a +$25$-vertex $5$-connected triangulation with degree sequence +$5^{21}\,8^{3}\,9^{1}$. At $n = 26$ there are again none. (All counts +depend on the correctness of \texttt{plantri} and of the Hamiltonicity +test.) + +An exhaustive scan of all $\binom{n}{5}$ candidate $5$-cuts confirms that +both $T$ and $T_{25}$ are \emph{internally $6$-connected}: every +$5$-element vertex cut is the neighbourhood of a single degree-$5$ +vertex, so neither admits a nontrivial separation of size $\le 5$. This +is the strongest connectivity a planar triangulation can have -- planar +graphs are never $6$-connected, because every planar graph has a vertex +of degree $\le 5$ -- and it is the level at which Birkhoff-style +reductions terminate (a minimal counterexample to the four colour theorem +can be assumed internally $6$-connected). Both cores are therefore +genuinely irreducible bases of any decomposition-based argument: nothing +in the cut-decomposition can simplify them further. + +The bridge-derivability test on $T_{25}$ follows the same recipe and +yields the same conclusion. Enumerating valid parity partitions and +ordering by total Betti, the minimum total Betti is $1$, and the very +first Betti-$1$ partition encountered with a small backward bridge-orbit +contains an Even Level Graph (source $s = 24$, maximum level $4$) at +depth $2$ in an orbit of $3{,}114$ states. The two bridge switches +carrying that Even Level Graph to $T_{25}$ are +\[ + \text{remove } \{21,23\},\ \text{add } \{22,24\} + \quad\text{and}\quad + \text{remove } \{3,5\},\ \text{add } \{1,6\}, +\] +each adding a same-parity edge that is a bridge of the even parity +subgraph; both steps are valid bridge switches +(Figure~\ref{fig:n25-dual}). The witness signature -- minimum total +Betti, a tiny bridge-orbit, a depth-$2$ Even Level Graph -- is the same +that worked on $T$, suggesting it is not an accident of the $n = 24$ +example but the generic shape of bridge-derivability witnesses on +internally-$6$-connected non-Hamiltonian-dual cores. The conjecture thus +survives every irreducible small case the connectivity bound forces us +to face. + +\begin{figure}[ht] +\centering +\includegraphics[width=0.7\textwidth]{figures/core_n25_dual.png} +\caption{The $25$-vertex dual $T_{25}$ of the unique $46$-vertex +non-Hamiltonian cyclically $5$-connected cubic planar graph -- the only +such cubic graph at $46$ vertices and the second internally +$6$-connected core known. Drawn crossing-free and coloured by parity +(blue even, orange odd) for its witness partition. $T_{25}$ is internally +$6$-connected and not an intertwining tree, yet is a bridge-derived level +graph: the two solid green edges $\{1,6\}$ and $\{22,24\}$ are the bridge +edges introduced by the two bridge switches carrying its witness Even +Level Graph (source $24$) to $T_{25}$. Each is a bridge of the even +parity subgraph.} +\label{fig:n25-dual} +\end{figure} + \subsection*{Toward a characterization of bridge-derived graphs} A bridge switch is a diagonal flip of the quadrilateral around a level