math-research/papers/face_monochromatic_pairs/experiments/draw_lemma_kempe_heawood.py

"""Two-panel illustration of the proof of Lemma 5.2
(Heawood constant on Kempe cycles through merged).

Each panel shows two consecutive vertices v_0, v_1 on the {a, b}-Kempe
cycle K, joined by an edge e, with h(v_0) = +1 (CW colour order (a, b, c))
and h(v_1) = -1 (CW colour order (a, c, b)).

Left panel (Case A): phi(e) = a. The two b-edges at v_0, v_1 both lie on
                     the same face F = F_R (right side of e); they form
                     the witness (e_1, e_2).

Right panel (Case B): phi(e) = b. The two a-edges at v_0, v_1 both lie
                      on the same face F = F_L (left side of e); they
                      form the witness (e_1, e_2).

Produces fig_lemma_kempe_heawood.png.
"""
import math
import os
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon

OUT_DIR = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))

DARK = '#374151'
GRAY = '#9ca3af'

# Colour code matching earlier figures: a=red/orange, b=blue, c=green.
COL_A = '#ea580c'   # 'a'
COL_B = '#2563eb'   # 'b'
COL_C = '#16a34a'   # 'c'

FACE_FILL = '#fef3c7'

V0 = (-1.6, 0.0)
V1 = ( 1.6, 0.0)


def edge_at(v, angle_deg, length=1.4):
    a = math.radians(angle_deg)
    return (v[0] + length * math.cos(a), v[1] + length * math.sin(a))


def draw_edge(ax, p, q, color, lw=2.6, zorder=2):
    ax.plot([p[0], q[0]], [p[1], q[1]], color=color, lw=lw,
            solid_capstyle='round', zorder=zorder)


def draw_vertex(ax, p, color=DARK, size=110, zorder=4):
    ax.scatter([p[0]], [p[1]], s=size, color=color, zorder=zorder)


def draw_stub(ax, p, color=DARK, size=45, zorder=4):
    ax.scatter([p[0]], [p[1]], s=size, color=color, zorder=zorder)


def label_text(ax, p, text, color=DARK, fontsize=12, dx=0, dy=0,
               weight='normal'):
    ax.text(p[0] + dx, p[1] + dy, text, ha='center', va='center',
            fontsize=fontsize, color=color, zorder=6, weight=weight,
            bbox=dict(boxstyle='round,pad=0.18', facecolor='white',
                      edgecolor='none', alpha=0.85))


def label_edge_midpoint(ax, p, q, text, color, fontsize=11, offset=(0, 0)):
    mid = ((p[0] + q[0]) / 2 + offset[0],
           (p[1] + q[1]) / 2 + offset[1])
    ax.text(mid[0], mid[1], text, ha='center', va='center',
            fontsize=fontsize, color=color, zorder=6,
            bbox=dict(boxstyle='round,pad=0.16', facecolor='white',
                      edgecolor='none', alpha=0.9))


def shade_face(ax, pts, color=FACE_FILL, alpha=0.7):
    poly = Polygon(pts, facecolor=color, edgecolor='none',
                   alpha=alpha, zorder=1)
    ax.add_patch(poly)


def panel_case_A(ax):
    # phi(e) = a. v_0 has CW order (a, b, c) starting from e at 0 deg:
    #   e (a) at 0 deg, b-edge at 300 deg (southeast), c-edge at 120 deg
    #   (northwest).
    # v_1 has CW order (a, c, b) starting from e at 180 deg:
    #   e (a) at 180 deg, c-edge at 60 deg (northeast), b-edge at 240 deg
    #   (south-southwest).
    e_color = COL_A
    # Other endpoints (stubs) of the non-e edges.
    b0 = edge_at(V0, -60)   # b-edge at v_0, southeast
    c0 = edge_at(V0, 120)   # c-edge at v_0, northwest
    c1 = edge_at(V1, 60)    # c-edge at v_1, northeast
    b1 = edge_at(V1, 240)   # b-edge at v_1, southwest

    # Shade F_R = south face: vertices roughly (b0, V0, V1, b1) plus a
    # closing polygon below.
    shade_face(ax, [V0, V1, b1, (b1[0] + 0.2, b1[1] - 0.6),
                    (b0[0] - 0.2, b0[1] - 0.6), b0])
    label_text(ax, ((V0[0] + V1[0]) / 2, -1.6), 'face $F$', color=DARK,
               fontsize=12, weight='bold')

    # Edges
    draw_edge(ax, V0, V1, e_color)                       # e (color a)
    draw_edge(ax, V0, b0, COL_B)                          # b-edge at v_0
    draw_edge(ax, V0, c0, COL_C)                          # c-edge at v_0
    draw_edge(ax, V1, c1, COL_C)                          # c-edge at v_1
    draw_edge(ax, V1, b1, COL_B)                          # b-edge at v_1

    # Vertices
    draw_vertex(ax, V0, DARK)
    draw_vertex(ax, V1, DARK)
    draw_stub(ax, b0); draw_stub(ax, c0)
    draw_stub(ax, c1); draw_stub(ax, b1)

    # Labels
    label_text(ax, V0, '$v_0$', dy=0.28, fontsize=12)
    label_text(ax, (V0[0] - 0.05, V0[1] - 0.28), '$h_\\varphi\\!=\\!+1$',
               color=DARK, fontsize=9)
    label_text(ax, V1, '$v_1$', dy=0.28, fontsize=12)
    label_text(ax, (V1[0] + 0.05, V1[1] - 0.28), '$h_\\varphi\\!=\\!-1$',
               color=DARK, fontsize=9)

    label_edge_midpoint(ax, V0, V1, '$e\\!=\\!a$', color=COL_A,
                        offset=(0, 0.16))
    label_edge_midpoint(ax, V0, b0, '$e_1\\!=\\!b$', color=COL_B,
                        offset=(-0.05, 0.05))
    label_edge_midpoint(ax, V0, c0, '$c$', color=COL_C,
                        offset=(0.05, 0))
    label_edge_midpoint(ax, V1, c1, '$c$', color=COL_C,
                        offset=(-0.05, 0))
    label_edge_midpoint(ax, V1, b1, '$e_2\\!=\\!b$', color=COL_B,
                        offset=(0.05, 0.05))

    ax.set_title('Case A: $\\varphi(e) = a$.  The two $b$-edges'
                 ' at $v_0, v_1$ lie on $\\partial F$',
                 fontsize=11, color=DARK, pad=10, fontweight='bold')


def panel_case_B(ax):
    # phi(e) = b. v_0 has CW order (a, b, c) with b = e at 0 deg:
    #   a-edge at 60 deg (northeast), e (b) at 0 deg, c-edge at 300 deg
    #   (southeast).
    # v_1 has CW order (a, c, b) with b = e at 180 deg:
    #   a-edge at 60 deg (northeast), c-edge at 300 deg (southeast), e
    #   (b) at 180 deg.
    e_color = COL_B
    a0 = edge_at(V0, 60)    # a-edge at v_0, northeast
    c0 = edge_at(V0, -60)   # c-edge at v_0, southeast
    a1 = edge_at(V1, 120)   # a-edge at v_1, northwest
    c1 = edge_at(V1, 240)   # c-edge at v_1, southwest

    # Shade F_L = north face: a0, V0, V1, a1, plus a closing polygon above.
    shade_face(ax, [V0, a0, (a0[0] - 0.2, a0[1] + 0.6),
                    (a1[0] + 0.2, a1[1] + 0.6), a1, V1])
    label_text(ax, ((V0[0] + V1[0]) / 2, 1.6), 'face $F$', color=DARK,
               fontsize=12, weight='bold')

    # Edges
    draw_edge(ax, V0, V1, e_color)
    draw_edge(ax, V0, a0, COL_A)
    draw_edge(ax, V0, c0, COL_C)
    draw_edge(ax, V1, a1, COL_A)
    draw_edge(ax, V1, c1, COL_C)

    # Vertices
    draw_vertex(ax, V0, DARK)
    draw_vertex(ax, V1, DARK)
    draw_stub(ax, a0); draw_stub(ax, c0)
    draw_stub(ax, a1); draw_stub(ax, c1)

    # Labels
    label_text(ax, V0, '$v_0$', dy=0.28, fontsize=12)
    label_text(ax, (V0[0] - 0.05, V0[1] - 0.28), '$h_\\varphi\\!=\\!+1$',
               color=DARK, fontsize=9)
    label_text(ax, V1, '$v_1$', dy=0.28, fontsize=12)
    label_text(ax, (V1[0] + 0.05, V1[1] - 0.28), '$h_\\varphi\\!=\\!-1$',
               color=DARK, fontsize=9)

    label_edge_midpoint(ax, V0, V1, '$e\\!=\\!b$', color=COL_B,
                        offset=(0, -0.18))
    label_edge_midpoint(ax, V0, a0, '$e_1\\!=\\!a$', color=COL_A,
                        offset=(-0.05, 0))
    label_edge_midpoint(ax, V0, c0, '$c$', color=COL_C,
                        offset=(0.05, 0))
    label_edge_midpoint(ax, V1, a1, '$e_2\\!=\\!a$', color=COL_A,
                        offset=(0.05, 0))
    label_edge_midpoint(ax, V1, c1, '$c$', color=COL_C,
                        offset=(-0.05, 0))

    ax.set_title('Case B: $\\varphi(e) = b$.  The two $a$-edges'
                 ' at $v_0, v_1$ lie on $\\partial F$',
                 fontsize=11, color=DARK, pad=10, fontweight='bold')


def main():
    plt.rcParams['text.usetex'] = False  # keep matplotlib defaults
    fig, axes = plt.subplots(1, 2, figsize=(13, 5.5))
    for ax in axes:
        ax.set_xlim(-3.5, 3.5)
        ax.set_ylim(-2.4, 2.4)
        ax.set_aspect('equal')
        ax.axis('off')
    panel_case_A(axes[0])
    panel_case_B(axes[1])
    plt.subplots_adjust(left=0.02, right=0.98, top=0.90, bottom=0.04,
                        wspace=0.05)
    out = os.path.join(OUT_DIR, 'fig_lemma_kempe_heawood.png')
    plt.savefig(out, dpi=180, bbox_inches='tight')
    print(f"wrote {out}")


if __name__ == '__main__':
    main()