coloring_nested_tire_graphs: empirical test of rainbow + König-lift on cut tires
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 <noreply@anthropic.com>
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"""Empirical test of the rainbow conjecture and König-lift conjecture
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on cut tires from the new (pendant-redefined) definition.
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For each cut tire in our Holton-McKay #0 example:
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- Build the cut tire as a graph (face boundary + labelled pendants).
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- Enumerate proper edge 3-colorings via Sage chromatic_polynomial
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on the line graph (or direct brute force for small tires).
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- Compute the projection support onto (out spoke colors, in spoke
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colors) jointly.
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Tests:
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Rainbow conjecture analog:
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For tires whose face boundary is a simple cycle (spoke-only),
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by Prop 1.13 of paper.tex, # colorings = 2^n + 2(-1)^n.
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Spoke colors at each face-boundary vertex are uniquely
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determined by the cycle coloring (the "third color"). So
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projection support has specific structure.
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Stronger rainbow: face boundary closed walk that's not simple
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(= theta-graph-like). Apply Prop's from rainbow_proof.tex.
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König-lift conjecture analog:
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For pairs of adjacent cut tires (T_d, T_{d+1}) sharing in-spokes
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of T_d with face-boundary edges of T_{d+1}, do the projection
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supports overlap on the shared edges?
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"""
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import os
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import sys
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from itertools import permutations, product
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from collections import defaultdict
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from sage.all import Graph
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from cut_depth_label import (
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parse_planar_code, HM_FILE, find_six_edge_cut,
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apply_procedure, compute_nice_layout,
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)
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from cut_tire import cut_tire_at
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def build_cut_tire_graph(tire, d):
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"""Build a Sage Graph for the cut tire (face boundary + pendants),
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returning (G, edge_labels) where edge_labels[e] ∈
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{'face', 'out', 'in'} classifies the edge."""
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G = Graph(multiedges=False, loops=False)
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edge_labels = {}
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for (u, v) in tire['face_edges']:
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G.add_edge(u, v)
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edge_labels[(min(u, v), max(u, v))] = 'face'
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# Add pendants
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next_id = max(set(tire['face_vertices_unique']) | {0}) + 1000
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pendant_map = {}
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for (v, pid) in tire['out_spokes']:
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nv = next_id; next_id += 1
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G.add_edge(v, nv)
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edge_labels[(min(v, nv), max(v, nv))] = 'out'
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pendant_map[pid] = (v, nv)
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for (v, pid) in tire['in_spokes']:
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nv = next_id; next_id += 1
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G.add_edge(v, nv)
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edge_labels[(min(v, nv), max(v, nv))] = 'in'
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pendant_map[pid] = (v, nv)
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return G, edge_labels, pendant_map
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def enumerate_proper_edge_3colorings(G):
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"""Enumerate proper edge 3-colorings by computing the line graph
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and its proper 3-vertex-colorings."""
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L = G.line_graph(labels=False)
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edges = list(G.edges(labels=False))
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edges_norm = [(min(u, v), max(u, v)) for (u, v) in edges]
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n_edges = len(edges_norm)
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# Map L's vertices to edge indices
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L_verts = list(L.vertices())
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vertex_to_edge_idx = {}
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for i, e in enumerate(edges):
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e_can = frozenset(e)
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for v in L_verts:
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if frozenset(v) == e_can:
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vertex_to_edge_idx[v] = i
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break
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# Brute force over 3^n_edges, filtering for proper-coloring constraint
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colorings = []
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for assignment in product((1, 2, 3), repeat=n_edges):
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# Check: at each vertex of G, all incident edges have distinct colors
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good = True
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for v in G.vertices():
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incident = [edges_norm.index((min(v, w), max(v, w)))
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for w in G.neighbors(v)]
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colors_at_v = [assignment[i] for i in incident]
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if len(set(colors_at_v)) != len(colors_at_v):
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good = False
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break
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if good:
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colorings.append(assignment)
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return edges_norm, colorings
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def cut_tire_analysis(tire, d):
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"""Analyze a single cut tire: enumerate proper 3-edge-colorings,
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compute spoke projections, check for S_3-orbit closure."""
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G, edge_labels, _ = build_cut_tire_graph(tire, d)
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n = G.size()
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if n > 18: # too slow
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return None
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edges_norm, colorings = enumerate_proper_edge_3colorings(G)
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n_colorings = len(colorings)
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# Projection: spoke (out + in) tuple
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out_edge_indices = [i for i, e in enumerate(edges_norm)
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if edge_labels[e] == 'out']
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in_edge_indices = [i for i, e in enumerate(edges_norm)
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if edge_labels[e] == 'in']
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proj = set()
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for c in colorings:
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sigma = (tuple(c[i] for i in out_edge_indices),
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tuple(c[i] for i in in_edge_indices))
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proj.add(sigma)
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# S_3-closure check
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s3_closed = True
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for sigma in proj:
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for pi in permutations((1, 2, 3)):
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mapped = (tuple(pi[c - 1] for c in sigma[0]),
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tuple(pi[c - 1] for c in sigma[1]))
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if mapped not in proj:
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s3_closed = False
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break
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if not s3_closed:
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break
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# Count S_3-orbits
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seen = set()
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orbits = []
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for sigma in sorted(proj):
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if sigma in seen:
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continue
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orbit = set()
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for pi in permutations((1, 2, 3)):
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mapped = (tuple(pi[c - 1] for c in sigma[0]),
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tuple(pi[c - 1] for c in sigma[1]))
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orbit.add(mapped)
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orbit_in_proj = orbit & proj
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orbits.append(sorted(orbit_in_proj))
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seen |= orbit_in_proj
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return {
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'n_edges': n,
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'n_colorings': n_colorings,
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'projection_size': len(proj),
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'S3_closed': s3_closed,
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'n_orbits': len(orbits),
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'orbit_sizes': sorted(len(o) for o in orbits),
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'projection': proj,
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'out_count': len(out_edge_indices),
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'in_count': len(in_edge_indices),
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}
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def konig_lift_test(t1, t2, d1, d2, analysis1, analysis2):
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"""For adjacent tires (t1 at depth d1, t2 at depth d2 = d1+1),
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test König-lift compatibility.
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Shared structure: in spokes of t1 correspond to a subset of
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face-boundary edges of t2 (those that are also incident to t1's
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face boundary).
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"""
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# For now: compute the intersection of projections restricted to
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# the in-spoke side of t1. (Full identification of shared edges
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# requires more careful bookkeeping; this is a first approximation.)
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if d2 != d1 + 1:
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return None
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in_projections_t1 = set(sigma[1] for sigma in analysis1['projection'])
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out_projections_t2 = set(sigma[0] for sigma in analysis2['projection'])
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# These don't have a natural bijection without more setup; instead
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# compare their sizes and S_3-symmetry.
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return {
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't1_in_size': len(in_projections_t1),
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't2_out_size': len(out_projections_t2),
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# The "shared boundary" between t1 and t2 isn't trivially
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# identifiable from this aggregated data; flag as needing
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# explicit shared-edge identification.
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}
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def main():
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gs = parse_planar_code(HM_FILE)
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G = gs[0]
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S, cut = find_six_edge_cut(G)
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base_pos = compute_nice_layout(G)
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S1 = frozenset(G.vertices()) - frozenset(S)
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H1, pos1, ed1, _, _, _ = apply_procedure(
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G, S1, cut, base_pos, '1',
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pendant_start_id=max(G.vertices()) + 1 + 6)
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print(f"G'_1 depths range: 0 to {max(ed1.values())}\n")
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all_tires = {} # (d, face_idx) -> tire dict + analysis
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print("=" * 80)
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print("Cut tire analysis (per tire)")
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print("=" * 80)
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for d in range(1, max(ed1.values()) + 1):
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tires = cut_tire_at(H1, ed1, d)
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for f_idx, tire in enumerate(tires):
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n_face = tire['face_length']
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n_out = len(tire['out_spokes'])
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n_in = len(tire['in_spokes'])
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n_edges = n_face + n_out + n_in
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print(f'\ndepth {d}, face {f_idx}: |face|={n_face}, '
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f'out={n_out}, in={n_in}, total edges={n_edges}')
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if n_edges > 18:
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print(f' ... too many edges to enumerate ({n_edges} > 18); skipping')
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continue
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analysis = cut_tire_analysis(tire, d)
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if analysis is None:
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continue
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print(f" Proper edge 3-colorings: {analysis['n_colorings']}")
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print(f" Projection (out, in) size: {analysis['projection_size']}")
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print(f" S_3-closed: {analysis['S3_closed']}")
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print(f" # S_3-orbits: {analysis['n_orbits']}, "
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f"orbit sizes: {analysis['orbit_sizes']}")
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all_tires[(d, f_idx)] = (tire, analysis)
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# Rainbow analysis: for each tire, check if its projection is
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# related to a specific structured set (perms-per-face etc.)
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print("\n" + "=" * 80)
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print("Rainbow conjecture-style check")
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print("=" * 80)
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for (d, f_idx), (tire, analysis) in all_tires.items():
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n_out = analysis['out_count']
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n_in = analysis['in_count']
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if n_out == 0 or n_in == 0:
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print(f'\ndepth {d}, face {f_idx}: trivial '
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f'(one side has 0 spokes), skipping rainbow check')
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continue
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# Expected: if both sides have 3 spokes each, check perms-per-face
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if n_out == 3 and n_in == 3:
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expected_size = 36 # perms × perms
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actual = analysis['projection_size']
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print(f'\ndepth {d}, face {f_idx}: out={n_out}, in={n_in}; '
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f'predicted perms-per-face size = {expected_size}, '
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f'actual = {actual}')
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if __name__ == '__main__':
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main()
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\relax
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\@writefile{toc}{\contentsline {paragraph}{What would test the rainbow conjecture properly.}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{Cut-tire K\"onig-lift analog (tentative).}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{What this would require to test.}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{What's actually closer to provable.}{3}{}\protected@file@percent }
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\documentclass[11pt]{article}
|
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\usepackage{amsmath,amssymb,amsthm}
|
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\usepackage{graphicx}
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\usepackage{geometry}
|
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\usepackage{booktabs}
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\geometry{margin=1in}
|
||||
|
||||
\title{Testing the rainbow and K\"onig-lift conjectures on cut tires}
|
||||
\author{}
|
||||
\date{}
|
||||
|
||||
\newtheorem*{obs}{Observation}
|
||||
|
||||
\begin{document}
|
||||
\maketitle
|
||||
|
||||
\section*{What was tested}
|
||||
|
||||
For the cut 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}
|
||||
Reference in New Issue
Block a user