From d93e8d137ab05f73ee27bcb44666a811d4efe2e6 Mon Sep 17 00:00:00 2001 From: didericis Date: Thu, 11 Jun 2026 21:04:59 -0400 Subject: [PATCH] Add up-tooth apex colour-sequence experiment over n=9 up-teeth sweep Enumerate Kempe-balanced 3-colourings of every M(T) with |A(T)|=9 and a fixed number m of up teeth, record the up-tooth apex colour sequence (cyclic order, mod colour permutation only), and group the M(T) by their set of unique sequences. Runs for m=3,4,5,6 with per-sequence notes and figures plus a summary atlas. Finding: realised sequences obey outer-face Kempe parity (all three colour-counts share m's parity). Distinct sequences grow 1/4/10/28 while M(T) count falls 23/29/18/7 across m=3..6. 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For a fixed annular +size ``n`` and a fixed number ``m`` of up teeth we: + + 1. generate every full medial tire graph M(T) with |A(T)| = n and exactly + ``m`` up teeth (one representative per dihedral symmetry class); + 2. enumerate the Kempe-balanced (``valid``) proper 3-colourings of each M(T) + and read off the colour sequence of the up-tooth apex vertices u0 < u1 < + ... in increasing annular-edge order (i.e. cyclic order around A(T)); + 3. reduce each up-tooth sequence modulo the six colour permutations -- but + NOT modulo the dihedral symmetry of the cycle -- to a canonical sequence; + 4. group the M(T) by the *set* of canonical up-tooth sequences they realise, + and report how many M(T) share each set. + +For every canonical up-tooth sequence we emit a markdown note listing every +M(T) that realises it together with every Kempe-balanced colouring (modulo +colour permutation) on that M(T) producing the sequence, and a figure drawing +those colourings. A summary note records the step-4 grouping. + +Run: python3 kempe_up_tooth_sequences.py --n 9 --m 4 +""" + +from __future__ import annotations + +import argparse +import math +import os +from collections import defaultdict + +from full_medial_tire_generator import FullMedialTireGraph, generate +from kempe_valid_colorings import ( + adjacency, + classify_colorings, + kempe_components, +) + +HERE = os.path.dirname(os.path.abspath(__file__)) + +Coloring = dict[str, int] +PALETTE = {0: "#e6550d", 1: "#3182bd", 2: "#31a354"} +PALETTE_NAME = {0: "orange", 1: "blue", 2: "green"} + + +def canonical_sequence(seq: tuple[int, ...]) -> tuple[int, ...]: + """Relabel a colour sequence by first appearance (mod colour permutation).""" + remap: dict[int, int] = {} + out = [] + for c in seq: + if c not in remap: + remap[c] = len(remap) + out.append(remap[c]) + return tuple(out) + + +def seq_str(seq: tuple[int, ...]) -> str: + return "".join(str(c) for c in seq) + + +def _parity_partitions(m: int) -> set[tuple[int, ...]]: + """Colour multisets (n0,n1,n2) of m up-tooth apexes admissible under the + outer-face Kempe-parity rule: all three counts share m's parity. Returned + as tuples of the nonzero parts (descending), matching the table format.""" + out: set[tuple[int, ...]] = set() + for a in range(m + 1): + for b in range(a + 1): + c = m - a - b + if c < 0 or c > b: + continue + if a % 2 == b % 2 == c % 2 == m % 2: + out.add(tuple(v for v in (a, b, c) if v > 0)) + return out + + +def up_tooth_sequence(graph: FullMedialTireGraph, coloring: Coloring) -> tuple[int, ...]: + """Colours of the up-tooth apexes in increasing annular-edge order.""" + return tuple(coloring[f"u{i}"] for i in graph.up_edges) + + +def graph_id(idx: int) -> str: + return f"G{idx:02d}" + + +def describe_graph(graph: FullMedialTireGraph) -> str: + bites = ",".join(f"({i},{j})" for i, j in sorted(graph.bites)) or "-" + return f"word={graph.tooth_word} bites={bites}" + + +# --------------------------------------------------------------------------- +# Data collection. +# --------------------------------------------------------------------------- + +class Experiment: + def __init__(self, n: int, m: int): + self.n = n + self.m = m + self.graphs: list[FullMedialTireGraph] = [ + g for g in generate(n, min_up_teeth=3, dedup=True) if len(g.up_edges) == m + ] + # per graph: list of (coloring, canonical up-tooth sequence) + self.colorings: list[list[tuple[Coloring, tuple[int, ...]]]] = [] + # per graph: set of canonical up-tooth sequences it realises + self.graph_seq_sets: list[frozenset[tuple[int, ...]]] = [] + # canonical sequence -> list of (graph_idx, coloring) + self.by_sequence: dict[tuple[int, ...], list[tuple[int, Coloring]]] = defaultdict(list) + + for gidx, g in enumerate(self.graphs): + entries: list[tuple[Coloring, tuple[int, ...]]] = [] + seqs: set[tuple[int, ...]] = set() + for coloring, verdict in classify_colorings(g, dedup_colors=True): + if not verdict.valid: + continue + cseq = canonical_sequence(up_tooth_sequence(g, coloring)) + entries.append((coloring, cseq)) + seqs.add(cseq) + self.by_sequence[cseq].append((gidx, coloring)) + self.colorings.append(entries) + self.graph_seq_sets.append(frozenset(seqs)) + + def groups(self): + """Map each set-of-sequences to the list of graph indices realising it.""" + groups: dict[frozenset[tuple[int, ...]], list[int]] = defaultdict(list) + for gidx, sset in enumerate(self.graph_seq_sets): + groups[sset].append(gidx) + # sort by descending membership, then by set size + return sorted(groups.items(), key=lambda kv: (-len(kv[1]), len(kv[0]))) + + def sequences(self) -> list[tuple[int, ...]]: + return sorted(self.by_sequence) + + +# --------------------------------------------------------------------------- +# Drawing (adapted from kempe_valid_colorings._draw_colored). +# --------------------------------------------------------------------------- + +def _positions(graph: FullMedialTireGraph) -> dict[str, tuple[float, float]]: + n = graph.n + + def ann_xy(k): + ang = math.pi / 2 - 2 * math.pi * k / n + return math.cos(ang), math.sin(ang) + + def mid_ang(i): + return math.pi / 2 - 2 * math.pi * (i + 0.5) / n + + pos = {f"a{k}": ann_xy(k) for k in range(n)} + matched = graph.bite_edges + for i, tooth in enumerate(graph.tooth_word): + if tooth == "U": + pos[f"u{i}"] = (1.42 * math.cos(mid_ang(i)), 1.42 * math.sin(mid_ang(i))) + elif i not in matched: + pos[f"d{i}"] = (0.58 * math.cos(mid_ang(i)), 0.58 * math.sin(mid_ang(i))) + for i, j in sorted(graph.bites): + corners = [ann_xy(i), ann_xy((i + 1) % n), ann_xy(j), ann_xy((j + 1) % n)] + cx = sum(p[0] for p in corners) / 4.0 + cy = sum(p[1] for p in corners) / 4.0 + pos[f"p{i}_{j}"] = (cx * 0.82, cy * 0.82) + return pos + + +def _draw(ax, graph, coloring, title): + pos = _positions(graph) + for u, v in graph.edges(): + ax.plot([pos[u][0], pos[v][0]], [pos[u][1], pos[v][1]], + color="#bbbbbb", lw=0.5, zorder=1) + for k in range(graph.n): + a, b = f"a{k}", f"a{(k + 1) % graph.n}" + ax.plot([pos[a][0], pos[b][0]], [pos[a][1], pos[b][1]], + color="#666666", lw=1.0, zorder=2) + for v, (x, y) in pos.items(): + is_bite = v.startswith("p") + ax.scatter([x], [y], s=34 if is_bite else 24, color=PALETTE[coloring[v]], + edgecolors="black", linewidths=0.5 if is_bite else 0.3, zorder=3) + # ring the up-tooth apexes whose colours form the recorded sequence + ux = [pos[f"u{i}"][0] for i in graph.up_edges] + uy = [pos[f"u{i}"][1] for i in graph.up_edges] + ax.scatter(ux, uy, s=120, facecolors="none", edgecolors="#222222", + linewidths=1.4, zorder=4) + ax.set_xlim(-1.65, 1.65) + ax.set_ylim(-1.85, 1.65) + ax.set_aspect("equal") + ax.axis("off") + ax.set_title(title, fontsize=6, pad=1.5) + + +def draw_sequence(exp: Experiment, seq, out_png, out_pdf): + import matplotlib + matplotlib.use("Agg") + import matplotlib.pyplot as plt + + entries = exp.by_sequence[seq] + cols = 10 + rows = math.ceil(len(entries) / cols) + fig, axes = plt.subplots(rows, cols, figsize=(cols * 1.5, rows * 1.7), squeeze=False) + for idx in range(rows * cols): + ax = axes[idx // cols][idx % cols] + if idx < len(entries): + gidx, coloring = entries[idx] + g = exp.graphs[gidx] + useq = seq_str(up_tooth_sequence(g, coloring)) + _draw(ax, g, coloring, f"{graph_id(gidx)} u={useq}") + else: + ax.axis("off") + fig.suptitle( + f"Kempe-balanced colourings with up-tooth apex sequence " + f"{seq_str(seq)} (mod colour permutation)\n" + f"n={exp.n}, m={exp.m} up teeth — {len(entries)} colourings on " + f"{len({g for g, _ in entries})} M(T); black rings mark up-tooth apexes", + fontsize=11, y=0.998, + ) + fig.tight_layout(rect=(0, 0, 1, 0.96)) + fig.savefig(out_png, dpi=170) + fig.savefig(out_pdf) + plt.close(fig) + print(f"wrote {out_png}") + + +# --------------------------------------------------------------------------- +# Markdown notes. +# --------------------------------------------------------------------------- + +def compact_coloring(graph: FullMedialTireGraph, coloring: Coloring) -> str: + """A readable one-line dump of the full colouring, grouped by vertex kind.""" + parts = [] + parts.append("A=" + "".join(str(coloring[f"a{k}"]) for k in range(graph.n))) + up = " ".join(f"u{i}:{coloring[f'u{i}']}" for i in graph.up_edges) + parts.append("U[" + up + "]") + downs = [] + for i in graph.singleton_down_edges: + downs.append(f"d{i}:{coloring[f'd{i}']}") + for i, j in sorted(graph.bites): + downs.append(f"p{i}_{j}:{coloring[f'p{i}_{j}']}") + if downs: + parts.append("D[" + " ".join(downs) + "]") + return " ".join(parts) + + +def write_sequence_note(exp: Experiment, seq, path, fig_name): + s = seq_str(seq) + # group entries by graph + by_graph: dict[int, list[Coloring]] = defaultdict(list) + for gidx, coloring in exp.by_sequence[seq]: + by_graph[gidx].append(coloring) + + colour_multiset = {} + for c in seq: + colour_multiset[c] = colour_multiset.get(c, 0) + 1 + counts = ", ".join(f"{n}×colour{c}" for c, n in sorted(colour_multiset.items())) + + lines = [] + lines.append(f"# Up-tooth apex sequence `{s}`") + lines.append("") + lines.append( + f"Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic " + f"order around A(T), reduced modulo the six colour permutations) for " + f"Kempe-balanced 3-colourings of M(T) with **n = {exp.n}**, " + f"**m = {exp.m} up teeth**." + ) + lines.append("") + lines.append(f"- Colour multiset: {counts}.") + lines.append(f"- Realised by **{len(by_graph)}** of {len(exp.graphs)} M(T) " + f"(dihedral classes).") + lines.append(f"- **{len(exp.by_sequence[seq])}** Kempe-balanced colourings " + f"(mod colour permutation) produce it.") + lines.append(f"- Figure: `{fig_name}` (black rings mark the up-tooth apexes).") + lines.append("") + lines.append("Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` " + "up-tooth apexes; `D[...]` singleton down apexes `d` and bite " + "apexes `p`. Colours are 0/1/2 = " + + ", ".join(f"{c}:{PALETTE_NAME[c]}" for c in (0, 1, 2)) + ".") + lines.append("") + for gidx in sorted(by_graph): + g = exp.graphs[gidx] + cols = by_graph[gidx] + lines.append(f"## {graph_id(gidx)} — {describe_graph(g)}") + lines.append("") + lines.append(f"{len(cols)} colouring(s) with up-tooth sequence `{s}`:") + lines.append("") + for coloring in cols: + raw = seq_str(up_tooth_sequence(g, coloring)) + lines.append(f"- up apexes (raw labels) `{raw}` → canonical `{s}` · " + f"`{compact_coloring(g, coloring)}`") + lines.append("") + with open(path, "w") as fh: + fh.write("\n".join(lines) + "\n") + print(f"wrote {path}") + + +def write_summary(exp: Experiment, path, notes_dir_name): + lines = [] + lines.append(f"# Up-tooth apex sequences of Kempe-balanced colourings " + f"(n={exp.n}, m={exp.m})") + lines.append("") + lines.append( + f"Every full medial tire graph M(T) with |A(T)| = {exp.n} and exactly " + f"{exp.m} up teeth, one representative per dihedral class: " + f"**{len(exp.graphs)} M(T)**. 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zoqD#@Adfg7h=r4z@srM@tOl!<&r?6kSQc}4RS)`c@DpZJnOBiYfu9X0$rIpPFZZal zUeTYS&LfK40S7B6h&~B>?TA!9emrO+evWD_pG62j=Dv_{|BON BxN-mh literal 0 HcmV?d00001 diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m4/seq_0110.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m4/seq_0110.md new file mode 100644 index 0000000..6468e9e --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m4/seq_0110.md @@ -0,0 +1,190 @@ +# Up-tooth apex sequence `0110` + +Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 4 up teeth**. + +- Colour multiset: 2×colour0, 2×colour1. +- Realised by **22** of 29 M(T) (dihedral classes). +- **69** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_0110.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G00 — word=UUUUDDDDD bites=- + +10 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:2 d6:1 d7:0 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:2 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101202 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:1 d7:1 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101212 U[u0:2 u1:0 u2:0 u3:2] D[d4:2 d5:0 d6:0 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012102021 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:1 d6:1 d7:0 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012102101 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:2 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:2 d7:1 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012102121 U[u0:2 u1:0 u2:0 u3:2] D[d4:1 d5:0 d6:0 d7:0 d8:2]` + +## G01 — word=UUUUDDDDD bites=(4,8) + +4 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u3:2] D[d5:2 d6:1 d7:0 p4_8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101201 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:1 d7:2 p4_8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012102012 U[u0:2 u1:0 u2:0 u3:2] D[d5:1 d6:2 d7:0 p4_8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012102102 U[u0:2 u1:0 u2:0 u3:2] D[d5:0 d6:2 d7:1 p4_8:1]` + +## G02 — word=UUUDUDDDD bites=- + +5 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:2 d6:1 d7:0 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:2 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101202 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:1 d7:1 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101212 U[u0:2 u1:0 u2:0 u4:2] D[d3:2 d5:0 d6:0 d7:0 d8:1]` + +## G03 — word=UUUDUDDDD bites=(3,5) + +2 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u4:2] D[d6:2 d7:0 d8:1 p3_5:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d6:1 d7:0 d8:2 p3_5:2]` + +## G04 — word=UUUDUDDDD bites=(3,8) + +2 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u4:2] D[d5:2 d6:1 d7:0 p3_8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101201 U[u0:2 u1:0 u2:0 u4:2] D[d5:0 d6:1 d7:2 p3_8:2]` + +## G05 — word=UUUDDUDDD bites=- + +7 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d4:2 d6:1 d7:0 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:2 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120102 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:2 d7:1 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120121 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:1 d6:0 d7:0 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121021 U[u0:2 u1:0 u2:0 u5:2] D[d3:0 d4:0 d6:1 d7:0 d8:2]` + +## G06 — word=UUUDDUDDD bites=(4,6) + +1 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101012 U[u0:2 u1:0 u2:0 u5:2] D[d3:2 d7:0 d8:1 p4_6:2]` + +## G07 — word=UUUDDUDDD bites=(3,8) + +1 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012101021 U[u0:2 u1:0 u2:0 u5:2] D[d4:2 d6:1 d7:0 p3_8:2]` + +## G10 — word=UUDUDUDDD bites=- + +5 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:2 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120102 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:2 d7:1 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120121 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:1 d6:0 d7:0 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d2:0 d4:0 d6:1 d7:0 d8:2]` + +## G12 — word=UUDUDUDDD bites=(2,4) + +2 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u3:0 u5:2] D[d6:2 d7:0 d8:1 p2_4:0]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121021 U[u0:2 u1:0 u3:0 u5:2] D[d6:1 d7:0 d8:2 p2_4:0]` + +## G14 — word=UUDUDDUDD bites=- + +3 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:2 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120102 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d5:2 d7:1 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:0 d5:2 d7:0 d8:1]` + +## G15 — word=UUDUDDUDD bites=(5,7) + +1 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u6:2] D[d2:0 d4:1 d8:2 p5_7:2]` + +## G16 — word=UUDUDDUDD bites=(2,4) + +1 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u3:0 u6:2] D[d5:2 d7:0 d8:1 p2_4:0]` + +## G18 — word=UUDUDDDUD bites=- + +3 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 d6:2 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:1 d6:1 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121201 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:0 d5:0 d6:1 d8:2]` + +## G19 — word=UUDUDDDUD bites=(6,8) + +1 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012120101 U[u0:2 u1:0 u3:0 u7:2] D[d2:0 d4:1 d5:2 p6_8:2]` + +## G21 — word=UUDDUUDDD bites=- + +4 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d3:1 d6:1 d7:0 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121021 U[u0:2 u1:0 u4:0 u5:2] D[d2:0 d3:0 d6:1 d7:0 d8:2]` + +## G22 — word=UUDDUUDDD bites=(3,6) + +1 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012021021 U[u0:2 u1:0 u4:0 u5:2] D[d2:1 d7:0 d8:2 p3_6:1]` + +## G23 — word=UUDDUDUDD bites=- + +5 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:1 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012012101 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:2 d8:2]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:2 d5:0 d7:1 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d2:1 d3:1 d5:2 d7:0 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u1:0 u4:0 u6:2] D[d2:0 d3:0 d5:2 d7:0 d8:1]` + +## G25 — word=UUDDUDUDD bites=(2,8) + +3 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2002` → canonical `0110` · `A=012012012 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:1 d7:0 p2_8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012012102 U[u0:2 u1:0 u4:0 u6:2] D[d3:2 d5:0 d7:1 p2_8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012021012 U[u0:2 u1:0 u4:0 u6:2] D[d3:1 d5:2 d7:0 p2_8:1]` + +## G26 — word=UDUDUDUDD bites=- + +6 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2112` → canonical `0110` · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:1 d7:0 d8:1]` +- up apexes (raw labels) `2112` → canonical `0110` · `A=010202101 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:2 d8:2]` +- up apexes (raw labels) `2112` → canonical `0110` · `A=010202102 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d3:1 d5:0 d7:1 d8:1]` +- up apexes (raw labels) `2112` → canonical `0110` · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:2 d8:2]` +- up apexes (raw labels) `2112` → canonical `0110` · `A=012020102 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d5:2 d7:1 d8:1]` +- up apexes (raw labels) `2002` → canonical `0110` · `A=012121012 U[u0:2 u2:0 u4:0 u6:2] D[d1:0 d3:0 d5:2 d7:0 d8:1]` + +## G27 — word=UDUDUDUDD bites=(5,7) + +1 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2112` → canonical `0110` · `A=012020101 U[u0:2 u2:1 u4:1 u6:2] D[d1:0 d3:1 d8:2 p5_7:2]` + +## G28 — word=UDUDUDUDD bites=(3,5) + +1 colouring(s) with up-tooth sequence `0110`: + +- up apexes (raw labels) `2112` → canonical `0110` · `A=010202012 U[u0:2 u2:1 u4:1 u6:2] D[d1:2 d7:0 d8:1 p3_5:1]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m4/seq_0110.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m4/seq_0110.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9f65cde4d7d6db8793234630eb4796c6f028bc1d GIT binary patch literal 218018 zcmbTd1zc6z`Yvn%3IZyvlp+m_P!Nz1N$GBoS|CWXR9XQ=x*G%}79}N(0@B?r-6=>4 zhF zfkR8t*b-@iqP_RmPYEZKlq$*yg`~av*99db6bk8JL(7BtiOZq-#K_db#+;V-?|*f$ zHBm*PXmz1kaY^V3(g{V&@yHsw@bK^di~aroN3{C?(jBhwteGV0* zqwO;X6BrMs|0=W`@<>w)qldOm&>|=N$HmFVNz2E>i{PWx$K?=%alylmwETbFENWw8 z3%|H9cmAs_F#r4y85H4+)chPuBiEbp2Hza=I1r{U~|i zEAyngnuia{@|cVI2lQ%F6sZ^35rg zb@~zEox=}#6J;946FNDoj*a74_EtPT>>j}$tZCLZT0#d%9d+cDqI^5wJ~Bv;uUPaFkR>g`D&{z 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z*nJYcjuSz~C4am8L9^pHPOmW;-Q^I{Y}YUg>Jt5vPfJ4$0trIWy<6LQqf$l`uNNw; zpJ#`HIAm1HS6gsOhS3d(t^ofNGxUGB+pSN09S*fx^bGKU1^6GvfbrN%m#K@#L}_SG z|7b{`DA)U6=2fDq$zW@LL literal 0 HcmV?d00001 diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01002.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01002.md new file mode 100644 index 0000000..9cee7d0 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01002.md @@ -0,0 +1,45 @@ +# Up-tooth apex sequence `01002` + +Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**. + +- Colour multiset: 3×colour0, 1×colour1, 1×colour2. +- Realised by **5** of 18 M(T) (dihedral classes). +- **9** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_01002.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G11 — word=UUDUUDUDD bites=- + +5 colouring(s) with up-tooth sequence `01002`: + +- up apexes (raw labels) `20221` → canonical `01002` · `A=012010201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:2 d8:2]` +- up apexes (raw labels) `20221` → canonical `01002` · `A=012010202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:1 d5:1 d7:1 d8:1]` +- up apexes (raw labels) `20221` → canonical `01002` · `A=012101021 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:2 d7:0 d8:2]` +- up apexes (raw labels) `20221` → canonical `01002` · `A=012101201 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:2 d8:2]` +- up apexes (raw labels) `20221` → canonical `01002` · `A=012101202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[d2:0 d5:0 d7:1 d8:1]` + +## G12 — word=UUDUUDUDD bites=(2,8),(5,7) + +1 colouring(s) with up-tooth sequence `01002`: + +- up apexes (raw labels) `20221` → canonical `01002` · `A=012010202 U[u0:2 u1:0 u3:2 u4:2 u6:1] D[p2_8:1 p5_7:1]` + +## G13 — word=UUDUDUUDD bites=- + +1 colouring(s) with up-tooth sequence `01002`: + +- up apexes (raw labels) `20221` → canonical `01002` · `A=012101021 U[u0:2 u1:0 u3:2 u5:2 u6:1] D[d2:0 d4:2 d7:0 d8:2]` + +## G15 — word=UUDUDUDUD bites=- + +1 colouring(s) with up-tooth sequence `01002`: + +- up apexes (raw labels) `20221` → canonical `01002` · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[d2:1 d4:2 d6:2 d8:1]` + +## G17 — word=UUDUDUDUD bites=(2,8),(4,6) + +1 colouring(s) with up-tooth sequence `01002`: + +- up apexes (raw labels) `20221` → canonical `01002` · `A=012010102 U[u0:2 u1:0 u3:2 u5:2 u7:1] D[p2_8:1 p4_6:2]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01002.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01002.pdf new file mode 100644 index 0000000000000000000000000000000000000000..24c7ed84f07d681e63859d9e35687d251aaf43e9 GIT binary patch literal 41232 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`U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G11 — word=UUDUUDUDD bites=- + +3 colouring(s) with up-tooth sequence `01020`: + +- up apexes (raw labels) `20212` → canonical `01020` · `A=012102012 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:1 d7:0 d8:1]` +- up apexes (raw labels) `20212` → canonical `01020` · `A=012102101 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:2 d8:2]` +- up apexes (raw labels) `20212` → canonical `01020` · `A=012102102 U[u0:2 u1:0 u3:2 u4:1 u6:2] D[d2:0 d5:0 d7:1 d8:1]` + +## G13 — word=UUDUDUUDD bites=- + +2 colouring(s) with up-tooth sequence `01020`: + +- up apexes (raw labels) `20212` → canonical `01020` · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:1 d4:0 d7:0 d8:1]` +- up apexes (raw labels) `20212` → canonical `01020` · `A=012102012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[d2:0 d4:1 d7:0 d8:1]` + +## G14 — word=UUDUDUUDD bites=(2,8),(4,7) + +1 colouring(s) with up-tooth sequence `01020`: + +- up apexes (raw labels) `20212` → canonical `01020` · `A=012012012 U[u0:2 u1:0 u3:2 u5:1 u6:2] D[p2_8:1 p4_7:0]` + +## G15 — word=UUDUDUDUD bites=- + +1 colouring(s) with up-tooth sequence `01020`: + +- up apexes (raw labels) `20212` → canonical `01020` · `A=012010201 U[u0:2 u1:0 u3:2 u5:1 u7:2] D[d2:1 d4:2 d6:1 d8:2]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01020.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01020.pdf new file mode 100644 index 0000000000000000000000000000000000000000..54a0a0811f1897bf443cc7c33beb4dcd9b2b58e0 GIT binary patch literal 35829 zcmaI71z42b);>;-#2^ULh;+x$AR*EX(hUO)-Jz69OLq#=Asy0!bayw>B}z!hi}By% zd(L;x`Og3QU0m11!?T|id#%0ib+0{)8ZvU6JY4)Zj1{Xi<*hh8G%y+$3wss|gL3q&cf*UkxBqVSU&Q{19h5v1UoQtz34d0&|xb@X7>=Blp zG(vxWmG$?xcr2g?P{f~D=8k2uhhv&ZyKt7`&)f(O@xPwm%Amf2dTd% z4Yw-7$__5&;teFiz#k7x7`*WEa`Drc;BZR=yTD)%8j-(me&X!x0>1Ge@BDXPApiYO z=cyo^Z9HwkGyWbTXXgmarQw!y1U`^KSh`ptkXn0qx+CCDINn)fZ;d9{hrF^+I?TG1 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colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**. + +- Colour multiset: 1×colour0, 3×colour1, 1×colour2. +- Realised by **8** of 18 M(T) (dihedral classes). +- **17** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_01121.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G04 — word=UUUDUUDDD bites=- + +3 colouring(s) with up-tooth sequence `01121`: + +- up apexes (raw labels) `20010` → canonical `01121` · `A=012102101 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:2 d8:2]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012102102 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u5:0] D[d3:2 d6:0 d7:0 d8:2]` + +## G05 — word=UUUDUDUDD bites=- + +3 colouring(s) with up-tooth sequence `01121`: + +- up apexes (raw labels) `20010` → canonical `01121` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:2 d7:0 d8:2]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[d3:0 d5:1 d7:0 d8:1]` + +## G06 — word=UUUDUDUDD bites=(3,8),(5,7) + +1 colouring(s) with up-tooth sequence `01121`: + +- up apexes (raw labels) `20010` → canonical `01121` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u6:0] D[p3_8:2 p5_7:0]` + +## G07 — word=UUUDUDDUD bites=- + +5 colouring(s) with up-tooth sequence `01121`: + +- up apexes (raw labels) `20010` → canonical `01121` · `A=012102012 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:2 d8:1]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012102021 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:1 d6:1 d8:2]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012102121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120121 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:2 d6:0 d8:2]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u2:0 u4:1 u7:0] D[d3:0 d5:1 d6:0 d8:1]` + +## G09 — word=UUUDDUUDD bites=- + +1 colouring(s) with up-tooth sequence `01121`: + +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u2:0 u5:1 u6:0] D[d3:0 d4:1 d7:0 d8:1]` + +## G11 — word=UUDUUDUDD bites=- + +2 colouring(s) with up-tooth sequence `01121`: + +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120121 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:2 d7:0 d8:2]` +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u3:0 u4:1 u6:0] D[d2:0 d5:1 d7:0 d8:1]` + +## G13 — word=UUDUDUUDD bites=- + +1 colouring(s) with up-tooth sequence `01121`: + +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u6:0] D[d2:0 d4:1 d7:0 d8:1]` + +## G15 — word=UUDUDUDUD bites=- + +1 colouring(s) with up-tooth sequence `01121`: + +- up apexes (raw labels) `20010` → canonical `01121` · `A=012120212 U[u0:2 u1:0 u3:0 u5:1 u7:0] D[d2:0 d4:1 d6:0 d8:1]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01121.pdf 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zt)E^QuWfMJ{nLNGe-bUKAKUQb+ujO8LhAcle|!a7ODX>EvH9^sF#E3kuSfak=YH(R q*`E8SGykhgpkMxfXQqnZgkI8bqh?kmX1f=_-|n46LptGT@V^1i9f-XE literal 0 HcmV?d00001 diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01200.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01200.md new file mode 100644 index 0000000..8e32eb2 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01200.md @@ -0,0 +1,125 @@ +# Up-tooth apex sequence `01200` + +Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**. + +- Colour multiset: 3×colour0, 1×colour1, 1×colour2. +- Realised by **15** of 18 M(T) (dihedral classes). +- **39** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_01200.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G00 — word=UUUUUDDDD bites=- + +6 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:2 d7:1 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:2 d6:0 d7:0 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010201 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010202 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:1 d7:1 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010212 U[u0:2 u1:0 u2:1 u3:2 u4:2] D[d5:1 d6:0 d7:0 d8:1]` + +## G01 — word=UUUUDUDDD bites=- + +3 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:2 d7:1 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010121 U[u0:2 u1:0 u2:1 u3:2 u5:2] D[d4:2 d6:0 d7:0 d8:2]` + +## G02 — word=UUUUDDUDD bites=- + +5 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:2 d5:2 d7:1 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012012012 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:1 d7:0 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012012101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012012102 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[d4:0 d5:0 d7:1 d8:1]` + +## G03 — word=UUUUDDUDD bites=(4,8),(5,7) + +1 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u3:2 u6:2] D[p4_8:2 p5_7:2]` + +## G04 — word=UUUDUUDDD bites=- + +3 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:2 d7:1 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010121 U[u0:2 u1:0 u2:1 u4:2 u5:2] D[d3:2 d6:0 d7:0 d8:2]` + +## G05 — word=UUUDUDUDD bites=- + +2 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[d3:2 d5:2 d7:1 d8:1]` + +## G06 — word=UUUDUDUDD bites=(3,8),(5,7) + +1 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u6:2] D[p3_8:2 p5_7:2]` + +## G07 — word=UUUDUDDUD bites=- + +2 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:2 d6:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010201 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[d3:2 d5:1 d6:1 d8:2]` + +## G08 — word=UUUDUDDUD bites=(3,5),(6,8) + +1 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u4:2 u7:2] D[p3_5:2 p6_8:2]` + +## G09 — word=UUUDDUUDD bites=- + +5 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:2 d4:2 d7:1 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012020101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012020102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:1 d7:1 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[d3:1 d4:0 d7:0 d8:1]` + +## G10 — word=UUUDDUUDD bites=(3,8),(4,7) + +3 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012010101 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[p3_8:2 p4_7:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012020102 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[p3_8:1 p4_7:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012021012 U[u0:2 u1:0 u2:1 u5:2 u6:2] D[p3_8:1 p4_7:0]` + +## G13 — word=UUDUDUUDD bites=- + +3 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:2 d8:2]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012020102 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:1 d7:1 d8:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[d2:1 d4:0 d7:0 d8:1]` + +## G14 — word=UUDUDUUDD bites=(2,8),(4,7) + +2 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012020102 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[p2_8:1 p4_7:1]` +- up apexes (raw labels) `20122` → canonical `01200` · `A=012021012 U[u0:2 u1:0 u3:1 u5:2 u6:2] D[p2_8:1 p4_7:0]` + +## G15 — word=UUDUDUDUD bites=- + +1 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → canonical `01200` · `A=012020101 U[u0:2 u1:0 u3:1 u5:2 u7:2] D[d2:1 d4:1 d6:2 d8:2]` + +## G16 — word=UUDUDUDUD bites=(2,4),(6,8) + +1 colouring(s) with up-tooth sequence `01200`: + +- up apexes (raw labels) `20122` → 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colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 5 up teeth**. + +- Colour multiset: 1×colour0, 3×colour1, 1×colour2. +- Realised by **12** of 18 M(T) (dihedral classes). +- **26** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_01211.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G04 — word=UUUDUUDDD bites=- + +6 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012101 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:2 d8:2]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012102 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:2 d7:1 d8:1]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:2 d6:0 d7:0 d8:2]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021201 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:2 d8:2]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021202 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:1 d7:1 d8:1]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u5:0] D[d3:1 d6:0 d7:0 d8:1]` + +## G05 — word=UUUDUDUDD bites=- + +2 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:2 d5:0 d7:0 d8:2]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[d3:1 d5:0 d7:0 d8:1]` + +## G06 — word=UUUDUDUDD bites=(3,8),(5,7) + +2 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[p3_8:2 p5_7:0]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u6:0] D[p3_8:1 p5_7:0]` + +## G07 — word=UUUDUDDUD bites=- + +6 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:2 d8:1]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:1 d6:1 d8:2]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:2 d5:0 d6:0 d8:2]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021012 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:2 d8:1]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021021 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:2 d6:1 d8:2]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u4:0 u7:0] D[d3:1 d5:0 d6:0 d8:1]` + +## G09 — word=UUUDDUUDD bites=- + +2 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:2 d4:0 d7:0 d8:2]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[d3:1 d4:0 d7:0 d8:1]` + +## G10 — word=UUUDDUUDD bites=(3,8),(4,7) + +2 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012012121 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[p3_8:2 p4_7:0]` +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u2:1 u5:0 u6:0] D[p3_8:1 p4_7:0]` + +## G11 — word=UUDUUDUDD bites=- + +1 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[d2:1 d5:0 d7:0 d8:1]` + +## G12 — word=UUDUUDUDD bites=(2,8),(5,7) + +1 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u4:0 u6:0] D[p2_8:1 p5_7:0]` + +## G13 — word=UUDUDUUDD bites=- + +1 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[d2:1 d4:0 d7:0 d8:1]` + +## G14 — word=UUDUDUUDD bites=(2,8),(4,7) + +1 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u6:0] D[p2_8:1 p4_7:0]` + +## G15 — word=UUDUDUDUD bites=- + +1 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[d2:1 d4:0 d6:0 d8:1]` + +## G17 — word=UUDUDUDUD bites=(2,8),(4,6) + +1 colouring(s) with up-tooth sequence `01211`: + +- up apexes (raw labels) `20100` → canonical `01211` · `A=012021212 U[u0:2 u1:0 u3:1 u5:0 u7:0] D[p2_8:1 p4_6:0]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01211.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m5/seq_01211.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e27a62fdecbe4c45254e76b2d9f55880319e673d GIT binary patch literal 91845 zcma&N1z42P);4Sb(jqNLNk}&W5(3iQHAs#$NJ^><64H&NbaxLa-QC??(xJk?N6&fR zbKd&?i|b*A;n~mHd+inXzSkNWrPmV7>?~YpGzD`M`L$^56l@f>MiyuS0u-!juJ51} ztYU_ahE}$w6s$^yrceh84se4Cg^&;$)W!sHBk#Wqkg&CJq~QE>1FM#zkpbVQ zKgC@fB~={_9ibFle?L$%baaH;+fZ;L{zPL{H8V6Zw=t#Q`Nx0lZH-l-jug5;>}v@y z6{w3N1*^0*n1bj(Ud8_LDovsHFVkWF?+ZZ88!>x(D45;f>9eXp9c-QKjlp^l^H-r@ zm4}*`8;aVx0Eukihn)cT 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Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G00 — word=UUUUUUDDD bites=- + +2 colouring(s) with up-tooth sequence `000011`: + +- up apexes (raw labels) `222211` → canonical `000011` · `A=010102012 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:2 d7:0 d8:1]` +- up apexes (raw labels) `222211` → canonical `000011` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u5:1] D[d6:1 d7:0 d8:2]` + +## G01 — word=UUUUUDUDD bites=- + +1 colouring(s) with up-tooth sequence `000011`: + +- up apexes (raw labels) `222211` → canonical `000011` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u4:1 u6:1] D[d5:1 d7:0 d8:2]` + +## G02 — word=UUUUDUUDD bites=- + +3 colouring(s) with up-tooth sequence `000011`: + +- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:2 d7:0 d8:1]` +- up apexes (raw labels) `222211` → canonical `000011` · `A=010102021 U[u0:2 u1:2 u2:2 u3:2 u5:1 u6:1] D[d4:1 d7:0 d8:2]` +- up apexes (raw labels) `222200` → canonical `000011` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u6:0] D[d4:1 d7:0 d8:2]` + +## G03 — word=UUUUDUDUD bites=- + +2 colouring(s) with up-tooth sequence `000011`: + +- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:2 d6:0 d8:1]` +- up apexes (raw labels) `222200` → canonical `000011` · `A=010102121 U[u0:2 u1:2 u2:2 u3:2 u5:0 u7:0] D[d4:1 d6:0 d8:2]` + +## G04 — word=UUUDUUUDD bites=- + +1 colouring(s) with up-tooth sequence `000011`: + +- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u6:0] D[d3:2 d7:0 d8:1]` + +## G05 — word=UUUDUUDUD bites=- + +1 colouring(s) with up-tooth sequence `000011`: + +- up apexes (raw labels) `222200` → canonical `000011` · `A=010101212 U[u0:2 u1:2 u2:2 u4:2 u5:0 u7:0] D[d3:2 d6:0 d8:1]` + +## G06 — word=UUDUUDUUD bites=- + +2 colouring(s) with up-tooth sequence `000011`: + +- up apexes (raw labels) `222211` → canonical `000011` · `A=010101202 U[u0:2 u1:2 u3:2 u4:2 u6:1 u7:1] 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o)A-+K0juTz$J)51UbBC=gfD=EgWh!)`Jbik$=}VmqwV?s0UarI!vFvP literal 0 HcmV?d00001 diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000101.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000101.md new file mode 100644 index 0000000..72bd1ce --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000101.md @@ -0,0 +1,39 @@ +# Up-tooth apex sequence `000101` + +Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**. + +- Colour multiset: 4×colour0, 2×colour1. +- Realised by **4** of 7 M(T) (dihedral classes). +- **8** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_000101.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G02 — word=UUUUDUUDD bites=- + +1 colouring(s) with up-tooth sequence `000101`: + +- up apexes (raw labels) `222020` → canonical `000101` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u6:0] D[d4:1 d7:0 d8:2]` + +## G03 — word=UUUUDUDUD bites=- + +3 colouring(s) with up-tooth sequence `000101`: + +- up apexes (raw labels) `222020` → canonical `000101` · `A=010120121 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:1 d6:0 d8:2]` +- up apexes (raw labels) `222020` → canonical `000101` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:2 d8:1]` +- up apexes (raw labels) `222020` → canonical `000101` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u5:2 u7:0] D[d4:0 d6:1 d8:2]` + +## G05 — word=UUUDUUDUD bites=- + +3 colouring(s) with up-tooth sequence `000101`: + +- up apexes (raw labels) `222121` → canonical `000101` · `A=010120102 U[u0:2 u1:2 u2:2 u4:1 u5:2 u7:1] D[d3:0 d6:2 d8:1]` +- up apexes (raw labels) `222020` → canonical `000101` · `A=010121012 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:2 d8:1]` +- up apexes (raw labels) `222020` → canonical `000101` · `A=010121021 U[u0:2 u1:2 u2:2 u4:0 u5:2 u7:0] D[d3:0 d6:1 d8:2]` + +## G06 — word=UUDUUDUUD bites=- + +1 colouring(s) with up-tooth sequence `000101`: + +- up apexes (raw labels) `222121` → canonical `000101` · `A=010102102 U[u0:2 u1:2 u3:2 u4:1 u6:2 u7:1] D[d2:2 d5:0 d8:1]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000101.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000101.pdf new file mode 100644 index 0000000000000000000000000000000000000000..1ec8adfdbb361eafb4daf9d651d1dfcd0a2166d6 GIT binary patch literal 39504 zcmbTd1z1$y)(1=|Al+S3qU2E0NP|iv(ls;;-Ju{zH%N=p-NOJ9(%qc`N+=;nBO>1( z|M%Ya{_p$U_j{i2;hC9p4riae_S(O-_HVDX4y(GH0vCi^0GqXZnX&8xHiVIv(aGEn 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z92`C@k5)f5uXJ?5cX43oRy}q5F=Br%br;{5yedU!|NsSOkgY& zE<6@}I&k&RAVN18{%0Qk*G6W4WC=2Q`~Ksz+aGoR_mP?Azx$tm4aogFll;GT=D)v= z{dtM+zr%QcO#SQeP|&}w%Wi>%;_(9q4k&Y6+w1EeBvaskAK(uz%y0h*JTmgHll|`x t%3*`l|NH#FF_wfJ# literal 0 HcmV?d00001 diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000110.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000110.md new file mode 100644 index 0000000..ae4c6d4 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000110.md @@ -0,0 +1,43 @@ +# Up-tooth apex sequence `000110` + +Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**. + +- Colour multiset: 4×colour0, 2×colour1. +- Realised by **5** of 7 M(T) (dihedral classes). +- **7** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_000110.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G00 — word=UUUUUUDDD bites=- + +2 colouring(s) with up-tooth sequence `000110`: + +- up apexes (raw labels) `222002` → canonical `000110` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:2 d7:0 d8:1]` +- up apexes (raw labels) `222002` → canonical `000110` · `A=010121021 U[u0:2 u1:2 u2:2 u3:0 u4:0 u5:2] D[d6:1 d7:0 d8:2]` + +## G01 — word=UUUUUDUDD bites=- + +1 colouring(s) with up-tooth sequence `000110`: + +- up apexes (raw labels) `222002` → canonical `000110` · `A=010121012 U[u0:2 u1:2 u2:2 u3:0 u4:0 u6:2] D[d5:2 d7:0 d8:1]` + +## G03 — word=UUUUDUDUD bites=- + +1 colouring(s) with up-tooth sequence `000110`: + +- up apexes (raw labels) `222002` → canonical `000110` · `A=010121201 U[u0:2 u1:2 u2:2 u3:0 u5:0 u7:2] D[d4:0 d6:1 d8:2]` + +## G04 — word=UUUDUUUDD bites=- + +1 colouring(s) with up-tooth sequence `000110`: + +- up apexes (raw labels) `222112` → canonical `000110` · `A=010102012 U[u0:2 u1:2 u2:2 u4:1 u5:1 u6:2] D[d3:2 d7:0 d8:1]` + +## G05 — word=UUUDUUDUD bites=- + +2 colouring(s) with up-tooth sequence `000110`: + +- up apexes (raw labels) `222112` → canonical `000110` · `A=010120201 U[u0:2 u1:2 u2:2 u4:1 u5:1 u7:2] D[d3:0 d6:1 d8:2]` +- up apexes (raw labels) `222002` → canonical `000110` · `A=010121201 U[u0:2 u1:2 u2:2 u4:0 u5:0 u7:2] D[d3:0 d6:1 d8:2]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000110.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_000110.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fb5f67dd3a25f7c8bc88c02d60438df705c3c664 GIT binary patch literal 36644 zcmagF1zc2L*Y`~;-7$c4H$w;th;#|kjl>K{N-8ZV-OP}RC`fmgARPk;O1HE~hm@ec zd;DMb_1y8?@AL8F#F?|t-g~XJ_xYXgT5vv8QV|pt5+~p+TVN?|ArNH|VR>TfL?9!> zB5dI24rdWowDGoadE&q#{Lsb$?!^KHC+M-r$q~TaVCWO2{!;^$CvM&>Vt-E%Hqo(l 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a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001100.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001100.md new file mode 100644 index 0000000..1047dd6 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001100.md @@ -0,0 +1,37 @@ +# Up-tooth apex sequence `001100` + +Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**. + +- Colour multiset: 4×colour0, 2×colour1. +- Realised by **4** of 7 M(T) (dihedral classes). +- **6** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_001100.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G00 — word=UUUUUUDDD bites=- + +2 colouring(s) with up-tooth sequence `001100`: + +- up apexes (raw labels) `221122` → canonical `001100` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:2 d7:0 d8:1]` +- up apexes (raw labels) `221122` → canonical `001100` · `A=010201021 U[u0:2 u1:2 u2:1 u3:1 u4:2 u5:2] D[d6:1 d7:0 d8:2]` + +## G01 — word=UUUUUDUDD bites=- + +1 colouring(s) with up-tooth sequence `001100`: + +- up apexes (raw labels) `221122` → canonical `001100` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u4:2 u6:2] D[d5:2 d7:0 d8:1]` + +## G02 — word=UUUUDUUDD bites=- + +1 colouring(s) with up-tooth sequence `001100`: + +- up apexes (raw labels) `221122` → canonical `001100` · `A=010201012 U[u0:2 u1:2 u2:1 u3:1 u5:2 u6:2] D[d4:2 d7:0 d8:1]` + +## G06 — word=UUDUUDUUD bites=- + +2 colouring(s) with up-tooth sequence `001100`: + +- up apexes (raw labels) `221122` → canonical `001100` · `A=010202101 U[u0:2 u1:2 u3:1 u4:1 u6:2 u7:2] D[d2:1 d5:0 d8:2]` +- up apexes (raw labels) `220022` → canonical `001100` · `A=010212101 U[u0:2 u1:2 u3:0 u4:0 u6:2 u7:2] D[d2:1 d5:0 d8:2]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001100.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001100.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3315f7af420cfff4a825f53e38d6a9fa1fd458da GIT binary patch literal 32635 zcmb5W1z1(lx;9EHC0)|BkcLGoAky94vFHw!lJ4%35M+ULBOpk33ew#oNJ$Fl9k}=Y z?>T3m^PhVko@X)FWQ_U6_`WgU@qUBBs46MV&dtG#!C1aTQ`UmPO~XawY;K1kEKI|x z>E&Wc!zp3rVdmg$O~a{bW^L(C0|gD#X+%XaES+HY4F&&02We*~4;r4o4LEgG%|@m0#xLq@~H%)^p~_a8S@%{)9T-JEFn?!RJiYS@^;Y@MuW`2YD=H)jhCOAi_Y zAXZWusAB2qLBlEM2viXN=WmIB{+6RL`VZ=G|GNSAdf#jBW(lOT#Jc0DK^6Y2ggByw}>@!_Csn5yLZkG{>NdZO}Dm z^|I=l%%e41ZDUq~Fk_fL3XGIXsmqV+j_Wa{TA`M7}SYlt}r`dl>c@^7y 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zya7>(nlRW+N$XFMGiw-Mff4jq5?AscC6{6@RH!ec95HY9=yJp>{rKT=y8O$?arXC{ z`j6j5b9fB>qucmje}#+h|Mu_x8J_xIk4?aUO#8nc=*Ryyl*0T&u>WH%BEQQ;Njd)G zlmEw)#Bi7q?vx-K^nd-wfB#X=@J!DS=lg&E9YJYWKL3a3^8a7n|LzU_fBe(qkk{Un YNMZ_f6%)}^*ag4N8kratVJ=7f55WE;KL7v# literal 0 HcmV?d00001 diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001111.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001111.md new file mode 100644 index 0000000..a3596e1 --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001111.md @@ -0,0 +1,43 @@ +# Up-tooth apex sequence `001111` + +Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**. + +- Colour multiset: 2×colour0, 4×colour1. +- Realised by **5** of 7 M(T) (dihedral classes). +- **7** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_001111.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G00 — word=UUUUUUDDD bites=- + +2 colouring(s) with up-tooth sequence `001111`: + +- up apexes (raw labels) `221111` → canonical `001111` · `A=010202012 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:2 d7:0 d8:1]` +- up apexes (raw labels) `221111` → canonical `001111` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u5:1] D[d6:1 d7:0 d8:2]` + +## G01 — word=UUUUUDUDD bites=- + +1 colouring(s) with up-tooth sequence `001111`: + +- up apexes (raw labels) `221111` → canonical `001111` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u4:1 u6:1] D[d5:1 d7:0 d8:2]` + +## G02 — word=UUUUDUUDD bites=- + +1 colouring(s) with up-tooth sequence `001111`: + +- up apexes (raw labels) `221111` → canonical `001111` · `A=010202021 U[u0:2 u1:2 u2:1 u3:1 u5:1 u6:1] D[d4:1 d7:0 d8:2]` + +## G04 — word=UUUDUUUDD bites=- + +1 colouring(s) with up-tooth sequence `001111`: + +- up apexes (raw labels) `221111` → canonical `001111` · `A=010202021 U[u0:2 u1:2 u2:1 u4:1 u5:1 u6:1] D[d3:1 d7:0 d8:2]` + +## G06 — word=UUDUUDUUD bites=- + +2 colouring(s) with up-tooth sequence `001111`: + +- up apexes (raw labels) `220000` → canonical `001111` · `A=010121212 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:2 d5:0 d8:1]` +- up apexes (raw labels) `220000` → canonical `001111` · `A=010212121 U[u0:2 u1:2 u3:0 u4:0 u6:0 u7:0] D[d2:1 d5:0 d8:2]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001111.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001111.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9e54c81fd1514a09b143b24f18e21594b31b4a09 GIT binary patch literal 36051 zcmaHS1z1#F*EXRjp@QT9(jhQI3?U#TA>G|IfOJc$AR%2t4HANMNT;AQLrbSLD$*b& z#=pnsdFy-s@4LLNan5k|S!=J@=ic{Pn^i;hA&ie(fPl4pk)f=GfRBNf!PUxvKunB* zN5{|2hJi=M(#z7x)sBHj!_v;ilK~EH&|;8~Ah2;kU~Uxtw*e1bUA!3h|J=Z1pl;=0 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zI78|KSbGe@RqDO)(HEy2FoIB(!;SDzR(S(EjuBn^B@WKt*lly_Fmb};lu_&8;R}zf l(ZIx#0+o8nODa0GKkT0htNh@(V=l%31fH&bF6*2UngC_@J(>Uj literal 0 HcmV?d00001 diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001122.md b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001122.md new file mode 100644 index 0000000..ce9c29b --- /dev/null +++ b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001122.md @@ -0,0 +1,44 @@ +# Up-tooth apex sequence `001122` + +Canonical up-tooth apex colour sequence (read u0 < u1 < ... in cyclic order around A(T), reduced modulo the six colour permutations) for Kempe-balanced 3-colourings of M(T) with **n = 9**, **m = 6 up teeth**. + +- Colour multiset: 2×colour0, 2×colour1, 2×colour2. +- Realised by **5** of 7 M(T) (dihedral classes). +- **8** Kempe-balanced colourings (mod colour permutation) produce it. +- Figure: `seq_001122.png` (black rings mark the up-tooth apexes). + +Colouring dump key: `A=` annular cycle a0..a_{n-1}; `U[...]` up-tooth apexes; `D[...]` singleton down apexes `d` and bite apexes `p`. Colours are 0/1/2 = 0:orange, 1:blue, 2:green. + +## G02 — word=UUUUDUUDD bites=- + +2 colouring(s) with up-tooth sequence `001122`: + +- up apexes (raw labels) `221100` → canonical `001122` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:2 d7:0 d8:1]` +- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u6:0] D[d4:1 d7:0 d8:2]` + +## G03 — word=UUUUDUDUD bites=- + +2 colouring(s) with up-tooth sequence `001122`: + +- up apexes (raw labels) `221100` → canonical `001122` · `A=010201212 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:2 d6:0 d8:1]` +- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u2:1 u3:1 u5:0 u7:0] D[d4:1 d6:0 d8:2]` + +## G04 — word=UUUDUUUDD bites=- + +1 colouring(s) with up-tooth sequence `001122`: + +- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u6:0] D[d3:1 d7:0 d8:2]` + +## G05 — word=UUUDUUDUD bites=- + +1 colouring(s) with up-tooth sequence `001122`: + +- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u2:1 u4:1 u5:0 u7:0] D[d3:1 d6:0 d8:2]` + +## G06 — word=UUDUUDUUD bites=- + +2 colouring(s) with up-tooth sequence `001122`: + +- up apexes (raw labels) `220011` → canonical `001122` · `A=010121202 U[u0:2 u1:2 u3:0 u4:0 u6:1 u7:1] D[d2:2 d5:0 d8:1]` +- up apexes (raw labels) `221100` → canonical `001122` · `A=010202121 U[u0:2 u1:2 u3:1 u4:1 u6:0 u7:0] D[d2:1 d5:0 d8:2]` + diff --git a/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001122.pdf b/papers/medial_tire_decompositions_of_plane_triangulations/experiments/kempe_up_tooth_sequences_n9_m6/seq_001122.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c59b42ab99498b1f11a8a072b5cf4f129ca4ace2 GIT binary patch literal 39703 zcmb5VbzD?I+doW$lz?=npmaA%qaYw4(zP@!-7P7d(w)*$OCw4w2uRm1D3VHx(xC64 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