didericis
037d987c7d
The previous statement "Heawood is constant on K through merged" was
strictly stronger than what the proof actually established without
Conjecture 5.3. Restate the lemma in the contrapositive direction:
If h_phi is constant on V(K), then no edge e in E(K) admits a face
F of G'^hat and edges e_1, e_2 on dF realising the clause-(3) arc
of Conjecture 5.1 at the endpoints of e.
Proof structure is mostly preserved (same F_R/F_L geometry, same case
split on phi(e) in {a, b}, same reading-off of cyclic colour orders).
The hypothesis "h_phi(v_0) != h_phi(v_1)" becomes "h_phi(v_0) =
h_phi(v_1)", which flips the conclusion: the same-coloured non-e
edges at v_0, v_1 land on opposite faces of e instead of the same
face. No dependency on Conjecture 5.3 or Theorem 4.X.
Redraw the figure to match the new lemma: both vertices labelled
h_phi = +1, both showing CW order (a, b, c), and the same-colour pair
(b-edges in Case A, a-edges in Case B) drawn on opposite sides of e.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
203 lines
7.7 KiB
Python
203 lines
7.7 KiB
Python
"""Two-panel illustration of the proof of Lemma 5.2 (a Heawood-constant
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Kempe cycle does not admit the clause-(3) arc of Conjecture 5.1).
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Each panel shows two consecutive vertices v_0, v_1 on the {a, b}-Kempe
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cycle K, joined by an edge e, with h(v_0) = h(v_1) = +1: i.e., both
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have the same clockwise colour order (a, b, c). The would-be witness
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edges (b-edges in Case A, a-edges in Case B) lie on opposite sides of
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e, so no face of the graph contains both of them.
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Left panel (Case A): phi(e) = a. The b-edges at v_0, v_1 are on
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opposite sides of e (one south, one north).
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Right panel (Case B): phi(e) = b. The a-edges at v_0, v_1 are on
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opposite sides of e.
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Produces fig_lemma_kempe_heawood.png.
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"""
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import math
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import os
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import matplotlib.pyplot as plt
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from matplotlib.patches import Polygon
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OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))
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DARK = '#374151'
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GRAY = '#9ca3af'
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COL_A = '#ea580c' # 'a'
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COL_B = '#2563eb' # 'b'
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COL_C = '#16a34a' # 'c'
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FACE_FILL_R = '#fef3c7' # F_R shading (south)
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FACE_FILL_L = '#dbeafe' # F_L shading (north)
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V0 = (-1.6, 0.0)
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V1 = ( 1.6, 0.0)
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def edge_at(v, angle_deg, length=1.4):
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a = math.radians(angle_deg)
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return (v[0] + length * math.cos(a), v[1] + length * math.sin(a))
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def draw_edge(ax, p, q, color, lw=2.6, zorder=2):
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ax.plot([p[0], q[0]], [p[1], q[1]], color=color, lw=lw,
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solid_capstyle='round', zorder=zorder)
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def draw_vertex(ax, p, color=DARK, size=110, zorder=4):
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ax.scatter([p[0]], [p[1]], s=size, color=color, zorder=zorder)
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def draw_stub(ax, p, color=DARK, size=45, zorder=4):
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ax.scatter([p[0]], [p[1]], s=size, color=color, zorder=zorder)
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def label_text(ax, p, text, color=DARK, fontsize=12, dx=0, dy=0,
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weight='normal'):
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ax.text(p[0] + dx, p[1] + dy, text, ha='center', va='center',
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fontsize=fontsize, color=color, zorder=6, weight=weight,
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bbox=dict(boxstyle='round,pad=0.18', facecolor='white',
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edgecolor='none', alpha=0.85))
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def label_edge_midpoint(ax, p, q, text, color, fontsize=10, offset=(0, 0)):
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mid = ((p[0] + q[0]) / 2 + offset[0],
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(p[1] + q[1]) / 2 + offset[1])
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ax.text(mid[0], mid[1], text, ha='center', va='center',
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fontsize=fontsize, color=color, zorder=6,
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bbox=dict(boxstyle='round,pad=0.16', facecolor='white',
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edgecolor='none', alpha=0.9))
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def shade_face(ax, pts, color, alpha=0.55):
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poly = Polygon(pts, facecolor=color, edgecolor='none',
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alpha=alpha, zorder=1)
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ax.add_patch(poly)
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def panel_case_A(ax):
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# Same Heawood: v_0 and v_1 both have CW order (a, b, c) with e = a.
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# v_0: e at 0 deg (east), b at 300 deg (south), c at 60 deg (north).
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# v_1: e at 180 deg (west), b at 90 deg (north), c at 270 deg (south).
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# The b-edges land on opposite sides of e (south of v_0, north of v_1).
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b0 = edge_at(V0, 300) # south of v_0
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c0 = edge_at(V0, 60) # north of v_0
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b1 = edge_at(V1, 90) # north of v_1
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c1 = edge_at(V1, 270) # south of v_1
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# Shade both F_R (south) and F_L (north) lightly.
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shade_face(ax, [V0, V1, c1, (c1[0] + 0.3, c1[1] - 0.6),
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(b0[0] - 0.3, b0[1] - 0.6), b0], color=FACE_FILL_R)
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shade_face(ax, [V0, c0, (c0[0] - 0.3, c0[1] + 0.6),
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(b1[0] + 0.3, b1[1] + 0.6), b1, V1], color=FACE_FILL_L)
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label_text(ax, ((V0[0] + V1[0]) / 2, -1.7), '$F_R$', color=DARK,
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fontsize=11, weight='bold')
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label_text(ax, ((V0[0] + V1[0]) / 2, 1.7), '$F_L$', color=DARK,
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fontsize=11, weight='bold')
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draw_edge(ax, V0, V1, COL_A)
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draw_edge(ax, V0, b0, COL_B)
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draw_edge(ax, V0, c0, COL_C)
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draw_edge(ax, V1, b1, COL_B)
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draw_edge(ax, V1, c1, COL_C)
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draw_vertex(ax, V0, DARK); draw_vertex(ax, V1, DARK)
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draw_stub(ax, b0); draw_stub(ax, c0)
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draw_stub(ax, b1); draw_stub(ax, c1)
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label_text(ax, V0, '$v_0$', dy=0.28, fontsize=12)
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label_text(ax, (V0[0] - 0.05, V0[1] - 0.28), '$h_\\varphi\\!=\\!+1$',
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color=DARK, fontsize=9)
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label_text(ax, V1, '$v_1$', dy=0.28, fontsize=12)
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label_text(ax, (V1[0] + 0.05, V1[1] - 0.28), '$h_\\varphi\\!=\\!+1$',
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color=DARK, fontsize=9)
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label_edge_midpoint(ax, V0, V1, '$e\\!=\\!a$', color=COL_A,
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offset=(0, 0.18))
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label_edge_midpoint(ax, V0, b0, '$b$', color=COL_B,
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offset=(-0.15, 0))
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label_edge_midpoint(ax, V0, c0, '$c$', color=COL_C,
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offset=(-0.15, 0))
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label_edge_midpoint(ax, V1, b1, '$b$', color=COL_B,
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offset=(0.15, 0))
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label_edge_midpoint(ax, V1, c1, '$c$', color=COL_C,
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offset=(0.15, 0))
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ax.set_title('Case A: $\\varphi(e) = a$. The two $b$-edges are on\n'
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'opposite sides of $e$ -- no common face',
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fontsize=11, color=DARK, pad=8, fontweight='bold')
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def panel_case_B(ax):
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# Same Heawood: v_0 and v_1 both have CW order (a, b, c) with e = b.
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# v_0: a at 60 deg (north), e (b) at 0 deg (east), c at 300 deg (south).
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# v_1: a at 270 deg (south), e (b) at 180 deg (west), c at 90 deg (north).
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# The a-edges land on opposite sides of e (north of v_0, south of v_1).
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a0 = edge_at(V0, 60) # north of v_0
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c0 = edge_at(V0, 300) # south of v_0
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a1 = edge_at(V1, 270) # south of v_1
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c1 = edge_at(V1, 90) # north of v_1
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shade_face(ax, [V0, V1, a1, (a1[0] + 0.3, a1[1] - 0.6),
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(c0[0] - 0.3, c0[1] - 0.6), c0], color=FACE_FILL_R)
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shade_face(ax, [V0, a0, (a0[0] - 0.3, a0[1] + 0.6),
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(c1[0] + 0.3, c1[1] + 0.6), c1, V1], color=FACE_FILL_L)
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label_text(ax, ((V0[0] + V1[0]) / 2, -1.7), '$F_R$', color=DARK,
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fontsize=11, weight='bold')
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label_text(ax, ((V0[0] + V1[0]) / 2, 1.7), '$F_L$', color=DARK,
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fontsize=11, weight='bold')
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draw_edge(ax, V0, V1, COL_B)
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draw_edge(ax, V0, a0, COL_A)
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draw_edge(ax, V0, c0, COL_C)
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draw_edge(ax, V1, a1, COL_A)
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draw_edge(ax, V1, c1, COL_C)
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draw_vertex(ax, V0, DARK); draw_vertex(ax, V1, DARK)
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draw_stub(ax, a0); draw_stub(ax, c0)
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draw_stub(ax, a1); draw_stub(ax, c1)
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label_text(ax, V0, '$v_0$', dy=0.28, fontsize=12)
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label_text(ax, (V0[0] - 0.05, V0[1] - 0.28), '$h_\\varphi\\!=\\!+1$',
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color=DARK, fontsize=9)
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label_text(ax, V1, '$v_1$', dy=0.28, fontsize=12)
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label_text(ax, (V1[0] + 0.05, V1[1] - 0.28), '$h_\\varphi\\!=\\!+1$',
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color=DARK, fontsize=9)
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label_edge_midpoint(ax, V0, V1, '$e\\!=\\!b$', color=COL_B,
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offset=(0, -0.18))
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label_edge_midpoint(ax, V0, a0, '$a$', color=COL_A,
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offset=(-0.15, 0))
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label_edge_midpoint(ax, V0, c0, '$c$', color=COL_C,
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offset=(-0.15, 0))
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label_edge_midpoint(ax, V1, a1, '$a$', color=COL_A,
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offset=(0.15, 0))
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label_edge_midpoint(ax, V1, c1, '$c$', color=COL_C,
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offset=(0.15, 0))
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ax.set_title('Case B: $\\varphi(e) = b$. The two $a$-edges are on\n'
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'opposite sides of $e$ -- no common face',
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fontsize=11, color=DARK, pad=8, fontweight='bold')
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def main():
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fig, axes = plt.subplots(1, 2, figsize=(13, 5.8))
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for ax in axes:
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ax.set_xlim(-3.5, 3.5)
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ax.set_ylim(-2.5, 2.5)
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ax.set_aspect('equal')
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ax.axis('off')
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panel_case_A(axes[0])
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panel_case_B(axes[1])
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plt.subplots_adjust(left=0.02, right=0.98, top=0.90, bottom=0.04,
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wspace=0.05)
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out = os.path.join(OUT_DIR, 'fig_lemma_kempe_heawood.png')
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plt.savefig(out, dpi=180, bbox_inches='tight')
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print(f"wrote {out}")
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if __name__ == '__main__':
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main()
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