From 702cbcecf7be33142fc8be91e482d57274559111 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 16:09:05 -0400 Subject: [PATCH] =?UTF-8?q?coloring=5Fnested=5Ftire=5Fgraphs:=20empirical?= =?UTF-8?q?=20test=20of=20rainbow=20+=20K=C3=B6nig-lift=20on=20cut=20tires?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit For each cut tire on G'_1 of Holton-McKay #0 (HM cut: |S|=10, matching 6-cut), brute-force enumerate proper edge 3-colorings, compute the joint (σ_out, σ_in) projection, and check S_3-closure and orbit decomposition. Results (8 cut tires analyzed, 2 too big or trivial): d face |f| out in |E| #col |π| S3-cl orbit sizes 1 0 12 5 0 17 96 93 yes [3, 6^15] 1 1 4 1 0 5 6 3 yes [3] 2 0 7 4 3 14 126 126 yes [6^21] 2 1 7 4 3 14 126 126 yes [6^21] 3 0-2 2 0 0 2 3 1 yes [1] 4 0 4 1 0 5 6 3 yes [3] 4 1 8 2 1 11 24 21 yes [3, 6^3] 5 1 2 0 0 2 3 1 yes [1] 6 0 12 3 2 17 96 93 yes [3, 6^15] 7 0 2 0 0 2 3 1 yes [1] Findings: 1. S_3-closure is universal (structural, expected). 2. Orbit sizes are always 3 (constant) or 6 (generic). 3. Non-trivial cut tires have rich projections (e.g. d=2 has 21 size-6 orbits = 126 elements; d=6 has 16 orbits). Neither conjecture is DIRECTLY testable on this example: - Rainbow conjecture requires antipodal-chord SP face boundary structure. Our cut tires' face boundaries don't naturally have this shape. - König-lift conjecture requires both sides give γ-face partitions on a shared γ. Cut tires at consecutive depths share data via in-spoke ↔ face-boundary-edge bijection, not via γ-face partitions. What CAN be observed: cut tire projections are LARGE and S_3- symmetric (substantially looser than the rainbow case's 36-element prediction). A "loose conjecture" would say π(T) ≥ c · 6 with c depending only on |E(T)|, derivable from Prop 1.13 in paper.tex. Files: experiments/cut_tire_test.py notes/cut_tire_conjecture_tests.tex (3 pages) Co-Authored-By: Claude Opus 4.7 --- .../experiments/cut_tire_test.py | 244 +++++++++++++++ .../notes/cut_tire_conjecture_tests.aux | 6 + .../notes/cut_tire_conjecture_tests.log | 294 ++++++++++++++++++ .../notes/cut_tire_conjecture_tests.pdf | Bin 0 -> 164867 bytes .../notes/cut_tire_conjecture_tests.tex | 200 ++++++++++++ 5 files changed, 744 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/experiments/cut_tire_test.py create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.log create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.tex diff --git a/papers/coloring_nested_tire_graphs/experiments/cut_tire_test.py b/papers/coloring_nested_tire_graphs/experiments/cut_tire_test.py new file mode 100644 index 0000000..29c1c86 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/cut_tire_test.py @@ -0,0 +1,244 @@ +"""Empirical test of the rainbow conjecture and König-lift conjecture +on cut tires from the new (pendant-redefined) definition. + +For each cut tire in our Holton-McKay #0 example: + - Build the cut tire as a graph (face boundary + labelled pendants). + - Enumerate proper edge 3-colorings via Sage chromatic_polynomial + on the line graph (or direct brute force for small tires). + - Compute the projection support onto (out spoke colors, in spoke + colors) jointly. + +Tests: + Rainbow conjecture analog: + For tires whose face boundary is a simple cycle (spoke-only), + by Prop 1.13 of paper.tex, # colorings = 2^n + 2(-1)^n. + Spoke colors at each face-boundary vertex are uniquely + determined by the cycle coloring (the "third color"). So + projection support has specific structure. + + Stronger rainbow: face boundary closed walk that's not simple + (= theta-graph-like). Apply Prop's from rainbow_proof.tex. + + König-lift conjecture analog: + For pairs of adjacent cut tires (T_d, T_{d+1}) sharing in-spokes + of T_d with face-boundary edges of T_{d+1}, do the projection + supports overlap on the shared edges? +""" +import os +import sys +from itertools import permutations, product +from collections import defaultdict + +from sage.all import Graph + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from cut_depth_label import ( + parse_planar_code, HM_FILE, find_six_edge_cut, + apply_procedure, compute_nice_layout, +) +from cut_tire import cut_tire_at + + +def build_cut_tire_graph(tire, d): + """Build a Sage Graph for the cut tire (face boundary + pendants), + returning (G, edge_labels) where edge_labels[e] ∈ + {'face', 'out', 'in'} classifies the edge.""" + G = Graph(multiedges=False, loops=False) + edge_labels = {} + for (u, v) in tire['face_edges']: + G.add_edge(u, v) + edge_labels[(min(u, v), max(u, v))] = 'face' + # Add pendants + next_id = max(set(tire['face_vertices_unique']) | {0}) + 1000 + pendant_map = {} + for (v, pid) in tire['out_spokes']: + nv = next_id; next_id += 1 + G.add_edge(v, nv) + edge_labels[(min(v, nv), max(v, nv))] = 'out' + pendant_map[pid] = (v, nv) + for (v, pid) in tire['in_spokes']: + nv = next_id; next_id += 1 + G.add_edge(v, nv) + edge_labels[(min(v, nv), max(v, nv))] = 'in' + pendant_map[pid] = (v, nv) + return G, edge_labels, pendant_map + + +def enumerate_proper_edge_3colorings(G): + """Enumerate proper edge 3-colorings by computing the line graph + and its proper 3-vertex-colorings.""" + L = G.line_graph(labels=False) + edges = list(G.edges(labels=False)) + edges_norm = [(min(u, v), max(u, v)) for (u, v) in edges] + n_edges = len(edges_norm) + # Map L's vertices to edge indices + L_verts = list(L.vertices()) + vertex_to_edge_idx = {} + for i, e in enumerate(edges): + e_can = frozenset(e) + for v in L_verts: + if frozenset(v) == e_can: + vertex_to_edge_idx[v] = i + break + # Brute force over 3^n_edges, filtering for proper-coloring constraint + colorings = [] + for assignment in product((1, 2, 3), repeat=n_edges): + # Check: at each vertex of G, all incident edges have distinct colors + good = True + for v in G.vertices(): + incident = [edges_norm.index((min(v, w), max(v, w))) + for w in G.neighbors(v)] + colors_at_v = [assignment[i] for i in incident] + if len(set(colors_at_v)) != len(colors_at_v): + good = False + break + if good: + colorings.append(assignment) + return edges_norm, colorings + + +def cut_tire_analysis(tire, d): + """Analyze a single cut tire: enumerate proper 3-edge-colorings, + compute spoke projections, check for S_3-orbit closure.""" + G, edge_labels, _ = build_cut_tire_graph(tire, d) + n = G.size() + if n > 18: # too slow + return None + edges_norm, colorings = enumerate_proper_edge_3colorings(G) + n_colorings = len(colorings) + # Projection: spoke (out + in) tuple + out_edge_indices = [i for i, e in enumerate(edges_norm) + if edge_labels[e] == 'out'] + in_edge_indices = [i for i, e in enumerate(edges_norm) + if edge_labels[e] == 'in'] + proj = set() + for c in colorings: + sigma = (tuple(c[i] for i in out_edge_indices), + tuple(c[i] for i in in_edge_indices)) + proj.add(sigma) + # S_3-closure check + s3_closed = True + for sigma in proj: + for pi in permutations((1, 2, 3)): + mapped = (tuple(pi[c - 1] for c in sigma[0]), + tuple(pi[c - 1] for c in sigma[1])) + if mapped not in proj: + s3_closed = False + break + if not s3_closed: + break + # Count S_3-orbits + seen = set() + orbits = [] + for sigma in sorted(proj): + if sigma in seen: + continue + orbit = set() + for pi in permutations((1, 2, 3)): + mapped = (tuple(pi[c - 1] for c in sigma[0]), + tuple(pi[c - 1] for c in sigma[1])) + orbit.add(mapped) + orbit_in_proj = orbit & proj + orbits.append(sorted(orbit_in_proj)) + seen |= orbit_in_proj + return { + 'n_edges': n, + 'n_colorings': n_colorings, + 'projection_size': len(proj), + 'S3_closed': s3_closed, + 'n_orbits': len(orbits), + 'orbit_sizes': sorted(len(o) for o in orbits), + 'projection': proj, + 'out_count': len(out_edge_indices), + 'in_count': len(in_edge_indices), + } + + +def konig_lift_test(t1, t2, d1, d2, analysis1, analysis2): + """For adjacent tires (t1 at depth d1, t2 at depth d2 = d1+1), + test König-lift compatibility. + + Shared structure: in spokes of t1 correspond to a subset of + face-boundary edges of t2 (those that are also incident to t1's + face boundary). + """ + # For now: compute the intersection of projections restricted to + # the in-spoke side of t1. (Full identification of shared edges + # requires more careful bookkeeping; this is a first approximation.) + if d2 != d1 + 1: + return None + in_projections_t1 = set(sigma[1] for sigma in analysis1['projection']) + out_projections_t2 = set(sigma[0] for sigma in analysis2['projection']) + # These don't have a natural bijection without more setup; instead + # compare their sizes and S_3-symmetry. + return { + 't1_in_size': len(in_projections_t1), + 't2_out_size': len(out_projections_t2), + # The "shared boundary" between t1 and t2 isn't trivially + # identifiable from this aggregated data; flag as needing + # explicit shared-edge identification. + } + + +def main(): + gs = parse_planar_code(HM_FILE) + G = gs[0] + S, cut = find_six_edge_cut(G) + base_pos = compute_nice_layout(G) + S1 = frozenset(G.vertices()) - frozenset(S) + H1, pos1, ed1, _, _, _ = apply_procedure( + G, S1, cut, base_pos, '1', + pendant_start_id=max(G.vertices()) + 1 + 6) + + print(f"G'_1 depths range: 0 to {max(ed1.values())}\n") + + all_tires = {} # (d, face_idx) -> tire dict + analysis + print("=" * 80) + print("Cut tire analysis (per tire)") + print("=" * 80) + for d in range(1, max(ed1.values()) + 1): + tires = cut_tire_at(H1, ed1, d) + for f_idx, tire in enumerate(tires): + n_face = tire['face_length'] + n_out = len(tire['out_spokes']) + n_in = len(tire['in_spokes']) + n_edges = n_face + n_out + n_in + print(f'\ndepth {d}, face {f_idx}: |face|={n_face}, ' + f'out={n_out}, in={n_in}, total edges={n_edges}') + if n_edges > 18: + print(f' ... too many edges to enumerate ({n_edges} > 18); skipping') + continue + analysis = cut_tire_analysis(tire, d) + if analysis is None: + continue + print(f" Proper edge 3-colorings: {analysis['n_colorings']}") + print(f" Projection (out, in) size: {analysis['projection_size']}") + print(f" S_3-closed: {analysis['S3_closed']}") + print(f" # S_3-orbits: {analysis['n_orbits']}, " + f"orbit sizes: {analysis['orbit_sizes']}") + all_tires[(d, f_idx)] = (tire, analysis) + + # Rainbow analysis: for each tire, check if its projection is + # related to a specific structured set (perms-per-face etc.) + print("\n" + "=" * 80) + print("Rainbow conjecture-style check") + print("=" * 80) + for (d, f_idx), (tire, analysis) in all_tires.items(): + n_out = analysis['out_count'] + n_in = analysis['in_count'] + if n_out == 0 or n_in == 0: + print(f'\ndepth {d}, face {f_idx}: trivial ' + f'(one side has 0 spokes), skipping rainbow check') + continue + # Expected: if both sides have 3 spokes each, check perms-per-face + if n_out == 3 and n_in == 3: + expected_size = 36 # perms × perms + actual = analysis['projection_size'] + print(f'\ndepth {d}, face {f_idx}: out={n_out}, in={n_in}; ' + f'predicted perms-per-face size = {expected_size}, ' + f'actual = {actual}') + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_conjecture_tests.aux 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tires arising on $G'_1$ of Holton-McKay graph \#0 (with +the matching $6$-edge cut from \texttt{cut\_depth\_label.tex}), under +the redefined cut tire definition (face boundary $+$ labelled +pendants at degree-$2$ vertices), I: +\begin{enumerate} + \item Built each cut tire as a graph (face boundary plus pendants). + \item Brute-force enumerated proper edge $3$-colorings. + \item Computed the projection support $\{(\sigma_{\mathrm{out}}, + \sigma_{\mathrm{in}}) : \chi \text{ proper}\}$. + \item Checked $S_3$-closure and orbit decomposition. +\end{enumerate} + +\section*{Results} + +\begin{center} +\small +\begin{tabular}{lr|rrrrr|r|cl} +\toprule +$d$ & face & $|f|$ & out & in & $|E|$ & $\#$col. & $|\pi|$ + & $S_3$-cl.\ & orbit sizes\\ +\midrule +$1$ & $0$ & $12$ & $5$ & $0$ & $17$ & $96$ & $93$ & Yes & $[3, 6^{15}]$\\ +$1$ & $1$ & $4$ & $1$ & $0$ & $5$ & $6$ & $3$ & Yes & $[3]$\\ +$2$ & $0$ & $7$ & $4$ & $3$ & $14$ & $126$ & $126$ & Yes & $[6^{21}]$\\ +$2$ & $1$ & $7$ & $4$ & $3$ & $14$ & $126$ & $126$ & Yes & $[6^{21}]$\\ +$3$ & $0$--$2$ & $2$ & $0$ & $0$ & $2$ & $3$ & $1$ & Yes & $[1]$\\ +$4$ & $0$ & $4$ & $1$ & $0$ & $5$ & $6$ & $3$ & Yes & $[3]$\\ +$4$ & $1$ & $8$ & $2$ & $1$ & $11$ & $24$ & $21$ & Yes & $[3, 6^3]$\\ +$5$ & $0$ & $14$ & $4$ & $2$ & $20$ & --- (too big) & --- & --- & ---\\ +$5$ & $1$ & $2$ & $0$ & $0$ & $2$ & $3$ & $1$ & Yes & $[1]$\\ +$6$ & $0$ & $12$ & $3$ & $2$ & $17$ & $96$ & $93$ & Yes & $[3, 6^{15}]$\\ +$7$ & $0$ & $2$ & $0$ & $0$ & $2$ & $3$ & $1$ & Yes & $[1]$\\ +\bottomrule +\end{tabular} +\end{center} + +\section*{Observations} + +\begin{obs}[Universal $S_3$-closure] +Every cut tire's projection $\pi(\sigma_{\mathrm{out}}, \sigma_{\mathrm{in}})$ +is closed under the diagonal $S_3$ action on the $3$ colors. This +is structural and expected: proper edge $3$-coloring is color- +symmetric. +\end{obs} + +\begin{obs}[Orbit sizes are $3$ or $6$ only] +Every $S_3$-orbit in every cut tire projection has size $3$ (the +constant-color orbit, when present) or $6$ (the generic orbit using +all $3$ colors). No size-$2$ orbits, which would correspond to +$\sigma$'s with non-trivial $S_3$ stabilizer; these don't occur. +\end{obs} + +\begin{obs}[Non-trivial cut tires have substantial projection support] +The two ``main'' cut tires at depth $2$ (face length $7$, $4$ out +$+\ 3$ in spokes) each have $126$ proper edge $3$-colorings, all +$126$ distinct in their joint $(\sigma_{\mathrm{out}}, \sigma_{\mathrm{in}})$ +projection. This is $21$ full $S_3$-orbits of size $6$. + +By contrast the cut tire at depth $1$ (face length $12$, $5$ out +$+\ 0$ in) has $96$ colorings with $93$ distinct projections +($16$ orbits, including one size-$3$ orbit and fifteen size-$6$); +the small drop $96 - 93 = 3$ corresponds to the constant orbit. +\end{obs} + +\section*{Why the rainbow conjecture is not directly testable here} + +The rainbow conjecture from \texttt{rainbow\_proof.tex} states: +for an antipodal-chord SP tire $T = (m_1, (0, m/2), \mathrm{SP})$ +with $m \in \{4, 6\}$ even and $m_1 \ge m - 1$, the inner-spoke +projection $\pi_D(\mathcal{C}(T))$ equals the perms-per-face set +$\mathcal{P}_m$ (size $36$). + +For the rainbow conjecture to apply to a cut tire, the cut tire's +face boundary structure would need to match the antipodal-chord SP +structure: a cycle of length $m$ with $r = 2$ ``O-face''-analogous +pieces, each containing $m/2$ boundary edges. + +The cut tires in our example have face boundaries of lengths +$2, 4, 7, 8, 12, 14$, none of which structurally matches the +$\theta(1, p, q)$-shape with the antipodal-chord SP convention. +So the rainbow conjecture cannot be directly checked on these cut +tires; what we can confirm is the weaker structural property +($S_3$-closure, size-$6$ orbit structure), which is universal. + +\paragraph{What would test the rainbow conjecture properly.} +Find a cut tire whose face boundary is a closed walk visiting each +vertex twice, structured as $\theta(1, p, q)$ in the partial-tire- +dual sense. Such cut tires can arise when $H_d$ has a +``pinch'' vertex (cut vertex with two faces sharing that vertex). +The example $H_1$ has $4$ revisited vertices in its length-$12$ +face boundary, suggesting bridge-like structure; explicit +identification of whether this matches $\theta(1, p, q)$ would be +the next step. + +\section*{Why the K\"onig-lift conjecture is not directly testable here} + +The K\"onig-lift conjecture from \texttt{worst\_case\_proof\_sketch.tex} +applies to pairs of adjacent SP tires $(T_1, T_2)$ sharing a cycle +$\gamma$ where \emph{both} sides give direct $\gamma$-face partitions +(both have chord(s) on $\gamma$). + +For cut tires at depths $d$ and $d + 1$: the ``shared'' structure +is the bijection $\{\text{in spokes of } T_d\} +\leftrightarrow \{\text{specific face boundary edges of } T_{d+1}\}$. +This is not the same as ``both tires give $\gamma$-face partitions'' +because $T_{d+1}$'s face boundary is not (in general) a $\gamma$-cycle +that $T_d$ also borders --- the cut tires sit in $H_d$ and $H_{d+1}$ +respectively, with different vertex sets. + +So the K\"onig-lift conjecture would need restatement for the +cut-tire chain. A correct restatement might say: + +\paragraph{Cut-tire K\"onig-lift analog (tentative).} +For each pair of adjacent cut tires $(T_d, T_{d+1})$ in a chain, the +bijection $\beta : \{\text{in spokes of } T_d\} \to +\{\text{face-boundary edges of } T_{d+1} \text{ adjacent to } V(f_d)\}$ +preserves a Latin structure: any Latin $\sigma$ on the shared +positions in $T_{d+1}$'s face boundary can be lifted to a proper +edge $3$-coloring of $T_d$ via $\beta$, and vice versa. + +\paragraph{What this would require to test.} +\begin{enumerate} + \item Explicit identification of the bijection $\beta$ for each + pair $(T_d, T_{d+1})$. This requires tracking the planar + embedding inheritance from $G'_i$ through to $H_d$ and + $H_{d+1}$. + \item Computing the Latin structure on each $T_{d+1}$'s face + boundary edges (analogous to the perms-per-face structure + on $\gamma$). + \item Comparing the projected supports under $\beta$. +\end{enumerate} + +None of this is automated in the present code. + +\section*{What can be concluded from the empirical data} + +\begin{enumerate} + \item \textbf{$S_3$-closure holds universally.} Every cut tire + projection is $S_3$-symmetric, with orbits of size $3$ or $6$. + This matches the partial-tire-dual data + (\texttt{orbit\_decomposition.tex}, Obs.\ ``S3-closed''). + \item \textbf{Non-trivial cut tires have rich projections.} At + depth $2$, the projection is $126 = 21 \cdot 6$ ($21$ full + $S_3$-orbits). At depth $6$, $93 = 1 \cdot 3 + 15 \cdot 6$ + ($16$ orbits). This is substantially more than the + rainbow's $36 = 6 \cdot 6$ specific orbit, so cut tires here + are \emph{looser} than the rainbow case --- chain pigeonhole + should be \emph{easier} when projections are large. + \item \textbf{Trivial cut tires (length-$2$ faces) contribute + nothing.} Most depths have at least one length-$2$ face in + $H_d$ (degenerate), which gives a trivial cut tire with no + spokes. These are not informative for chain pigeonhole. + \item \textbf{Neither conjecture is directly applicable.} The + rainbow conjecture requires antipodal-chord SP structure, + which our cut tires don't naturally have. The K\"onig-lift + conjecture requires both sides give $\gamma$-face + partitions, which the cut-tire chain doesn't naturally + produce. +\end{enumerate} + +\paragraph{What's actually closer to provable.} +The empirical data suggests the projections are LARGE (and +$S_3$-symmetric) rather than SMALL. A natural chain pigeonhole +statement is: + +\begin{quote} +\textbf{Conjecture (loose).} For each cut tire $T$ arising in the +chain, the projection $\pi(T)$ has size $\ge c \cdot 6$ for some +absolute constant $c$ depending only on $|E(T)|$, with $\pi(T)$ +$S_3$-closed. Chain composition through $T_1, T_2, \ldots$ yields +$|\mathcal{R}_i| \ge c'$ uniform in chain length. +\end{quote} + +The constant $c$ might be derivable from Prop 1.13 of \texttt{paper.tex} +($2^n + 2(-1)^n$ colorings for spoke-only $D(T)$). An honest +chain pigeonhole then asks: do the cut-tire chain compositions on +both sides of the cut produce sufficient overlap to force +$\mathcal{R}_0 \cap \mathcal{R}_1 \neq \emptyset$? + +This requires the full chain machinery, not just per-tire analysis. + +\end{document}