face_monochromatic_pairs: Heawood numbers, Lemma 5.2 + diagram
- Add Definition 3.1 "Heawood number of a vertex" (+1 if CW colour order is (1,2,3), -1 if (1,3,2)) and cite Heawood 1898 in the bibliography. - Add Lemma 5.2 "Heawood number is constant on the Kempe cycles through the merged edge", positioned immediately after Conjecture 5.1. Its proof exhibits a (F, e_1, e_2) witness for clauses (1)-(3) of the conjecture from any pair (v_0, v_1) of consecutive K-vertices with differing Heawood signs, by cases on whether phi(e) = a or b. The proof does not invoke Conjecture 5.3 or Theorem 4.X. - Add a two-panel figure illustrating Case A (b-edges on F_R when phi(e) = a) and Case B (a-edges on F_L when phi(e) = b), with the cyclic colour orders (a, b, c) at v_0 and (a, c, b) at v_1 visible from the angular layout. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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"""Two-panel illustration of the proof of Lemma 5.2
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(Heawood constant on Kempe cycles through merged).
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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) = +1 (CW colour order (a, b, c))
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and h(v_1) = -1 (CW colour order (a, c, b)).
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Left panel (Case A): phi(e) = a. The two b-edges at v_0, v_1 both lie on
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the same face F = F_R (right side of e); they form
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the witness (e_1, e_2).
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Right panel (Case B): phi(e) = b. The two a-edges at v_0, v_1 both lie
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on the same face F = F_L (left side of e); they
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form the witness (e_1, e_2).
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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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# Colour code matching earlier figures: a=red/orange, b=blue, c=green.
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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 = '#fef3c7'
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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=11, 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=FACE_FILL, alpha=0.7):
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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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# phi(e) = a. v_0 has CW order (a, b, c) starting from e at 0 deg:
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# e (a) at 0 deg, b-edge at 300 deg (southeast), c-edge at 120 deg
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# (northwest).
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# v_1 has CW order (a, c, b) starting from e at 180 deg:
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# e (a) at 180 deg, c-edge at 60 deg (northeast), b-edge at 240 deg
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# (south-southwest).
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e_color = COL_A
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# Other endpoints (stubs) of the non-e edges.
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b0 = edge_at(V0, -60) # b-edge at v_0, southeast
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c0 = edge_at(V0, 120) # c-edge at v_0, northwest
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c1 = edge_at(V1, 60) # c-edge at v_1, northeast
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b1 = edge_at(V1, 240) # b-edge at v_1, southwest
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# Shade F_R = south face: vertices roughly (b0, V0, V1, b1) plus a
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# closing polygon below.
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shade_face(ax, [V0, V1, b1, (b1[0] + 0.2, b1[1] - 0.6),
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(b0[0] - 0.2, b0[1] - 0.6), b0])
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label_text(ax, ((V0[0] + V1[0]) / 2, -1.6), 'face $F$', color=DARK,
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fontsize=12, weight='bold')
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# Edges
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draw_edge(ax, V0, V1, e_color) # e (color a)
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draw_edge(ax, V0, b0, COL_B) # b-edge at v_0
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draw_edge(ax, V0, c0, COL_C) # c-edge at v_0
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draw_edge(ax, V1, c1, COL_C) # c-edge at v_1
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draw_edge(ax, V1, b1, COL_B) # b-edge at v_1
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# Vertices
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draw_vertex(ax, V0, DARK)
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draw_vertex(ax, V1, DARK)
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draw_stub(ax, b0); draw_stub(ax, c0)
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draw_stub(ax, c1); draw_stub(ax, b1)
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# Labels
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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, 0.16))
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label_edge_midpoint(ax, V0, b0, '$e_1\\!=\\!b$', color=COL_B,
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offset=(-0.05, 0.05))
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label_edge_midpoint(ax, V0, c0, '$c$', color=COL_C,
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offset=(0.05, 0))
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label_edge_midpoint(ax, V1, c1, '$c$', color=COL_C,
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offset=(-0.05, 0))
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label_edge_midpoint(ax, V1, b1, '$e_2\\!=\\!b$', color=COL_B,
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offset=(0.05, 0.05))
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ax.set_title('Case A: $\\varphi(e) = a$. The two $b$-edges'
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' at $v_0, v_1$ lie on $\\partial F$',
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fontsize=11, color=DARK, pad=10, fontweight='bold')
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def panel_case_B(ax):
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# phi(e) = b. v_0 has CW order (a, b, c) with b = e at 0 deg:
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# a-edge at 60 deg (northeast), e (b) at 0 deg, c-edge at 300 deg
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# (southeast).
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# v_1 has CW order (a, c, b) with b = e at 180 deg:
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# a-edge at 60 deg (northeast), c-edge at 300 deg (southeast), e
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# (b) at 180 deg.
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e_color = COL_B
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a0 = edge_at(V0, 60) # a-edge at v_0, northeast
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c0 = edge_at(V0, -60) # c-edge at v_0, southeast
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a1 = edge_at(V1, 120) # a-edge at v_1, northwest
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c1 = edge_at(V1, 240) # c-edge at v_1, southwest
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# Shade F_L = north face: a0, V0, V1, a1, plus a closing polygon above.
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shade_face(ax, [V0, a0, (a0[0] - 0.2, a0[1] + 0.6),
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(a1[0] + 0.2, a1[1] + 0.6), a1, V1])
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label_text(ax, ((V0[0] + V1[0]) / 2, 1.6), 'face $F$', color=DARK,
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fontsize=12, weight='bold')
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# Edges
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draw_edge(ax, V0, V1, e_color)
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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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# Vertices
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draw_vertex(ax, V0, DARK)
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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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# Labels
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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, '$e_1\\!=\\!a$', color=COL_A,
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offset=(-0.05, 0))
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label_edge_midpoint(ax, V0, c0, '$c$', color=COL_C,
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offset=(0.05, 0))
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label_edge_midpoint(ax, V1, a1, '$e_2\\!=\\!a$', color=COL_A,
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offset=(0.05, 0))
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label_edge_midpoint(ax, V1, c1, '$c$', color=COL_C,
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offset=(-0.05, 0))
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ax.set_title('Case B: $\\varphi(e) = b$. The two $a$-edges'
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' at $v_0, v_1$ lie on $\\partial F$',
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fontsize=11, color=DARK, pad=10, fontweight='bold')
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def main():
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plt.rcParams['text.usetex'] = False # keep matplotlib defaults
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fig, axes = plt.subplots(1, 2, figsize=(13, 5.5))
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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(-2.4, 2.4)
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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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