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face_monochromatic_pairs: Heawood numbers, Lemma 5.2 + diagram

Browse Source 
- 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>
This commit is contained in:
didericis
2026-05-24 21:54:30 -04:00
parent 41227c6a0f
commit d99f8e23b3
6 changed files with 391 additions and 63 deletions
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papers/face_monochromatic_pairs/experiments/draw_lemma_kempe_heawood.py
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"""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()
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papers/face_monochromatic_pairs/paper.aux
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@@ -11,41 +11,47 @@
\newlabel{sec:minimal}{{2}{2}}
\newlabel{lem:triangulate}{{2.1}{2}}
\newlabel{def:minimal}{{2.2}{2}}
\citation{Heawood1898}
\newlabel{lem:mindeg}{{2.4}{3}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The reduced dual}}{3}{}\protected@file@percent }
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.1\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{4}{}\protected@file@percent }
\newlabel{fig:reduced-dual-steps}{{1}{4}}
\newlabel{def:edge-names}{{3.3}{4}}
\newlabel{lem:pentagonal-externals}{{3.4}{5}}
\newlabel{lem:chord-apex}{{3.6}{6}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.6\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 3.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{7}{}\protected@file@percent }
\newlabel{def:heawood-number}{{3.1}{3}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 3.2\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{5}{}\protected@file@percent }
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\newlabel{lem:chord-apex}{{3.7}{6}}
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 3.7\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 3.5\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{7}{}\protected@file@percent }
\newlabel{fig:chord-apex-proof}{{2}{7}}
\newlabel{lem:kempe-spike}{{3.7}{7}}
\newlabel{lem:kempe-spike}{{3.8}{8}}
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\newlabel{thm:edge-suppression-4face}{{4.2}{8}}
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Edge suppression (Definition\nonbreakingspace 4.1\hbox {}). Left: a fragment of a cubic plane graph with the suppressed edge $e = uv$ highlighted in red. Middle: deleting $e$ leaves $u$ and $v$ of degree\nonbreakingspace $2$. Right: smoothing $u$ and $v$ replaces each pair of incident edges by a single new edge, removing $u, v$ and giving a cubic plane graph again.}}{9}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{The face-monochromatic-pair conjecture and the Four Colour Theorem}}{9}{}\protected@file@percent }
\newlabel{sec:toward-4ct}{{5}{9}}
\newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{9}}
\newlabel{thm:edge-suppression-4face}{{4.2}{9}}
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The recolouring used in the proof of Theorem\nonbreakingspace 4.2\hbox {}. Left: the $4$-face $f$ of $H$ under $\varphi $, with the forced colours $\varphi (e_0) = a$, $\varphi (e_1) = b$, $\varphi (e_2) = \varphi (e_3) = c$, $\varphi (w_0) = \varphi (w_1) = b$, and $\varphi (w_2) = \varphi (w_3) = a$. Right: the suppressed graph $H'$ under $\varphi '$. The smoothed-in edges $e_2', e_3'$ inherit the colour $b$ from $w_0, w_1$, and $e_1$ is recoloured from $b$ to $c$; every edge outside the face neighbourhood keeps its $\varphi $-colour (dotted in red: the five edges of $H$ removed by the suppression).}}{10}{}\protected@file@percent }
\newlabel{fig:thm-edge-suppression-4face}{{4}{10}}
\newlabel{rem:conj-3-6-empirical}{{5.2}{10}}
\newlabel{conj:face-monochromatic-pair-strengthened}{{5.3}{11}}
\newlabel{rem:conj-3-8-empirical}{{5.4}{11}}
\bibcite{AH77a}{1}
\bibcite{AHK77}{2}
\bibcite{RSST97}{3}
\bibcite{Gonthier08}{4}
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\newlabel{lem:kempe-heawood-constant}{{5.2}{11}}
\newlabel{rem:conj-3-6-empirical}{{5.3}{11}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The two cases in the proof of Lemma\nonbreakingspace 5.2\hbox {}. Vertices $v_0, v_1$ are consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$, with $h_\varphi (v_0) = +1$ (clockwise colour order $(a, b, c)$) and $h_\varphi (v_1) = -1$ (clockwise order $(a, c, b)$). \emph {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the two $b$-edges at $v_0, v_1$ lie on the same face $F$, with $e$ as the unique $\partial F$-edge between them. \emph {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the two $a$-edges at $v_0, v_1$ lie on the opposite face $F$ instead, again with $e$ between them on one arc. In either case $(F, e_1, e_2)$ witnesses clauses (1)--(3) of Conjecture\nonbreakingspace 5.1\hbox {}.}}{12}{}\protected@file@percent }
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\bibcite{AHK77}{3}
\bibcite{RSST97}{4}
\bibcite{Gonthier08}{5}
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papers/face_monochromatic_pairs/paper.log
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 15:03
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 21:49
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
@@ -195,102 +195,103 @@ Overfull \vbox (1.31958pt too high) has occurred while \output is active []
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[2]
Overfull \hbox (41.917pt too wide) in paragraph at lines 233--235
[2] [3]
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[]\OT1/cmr/m/n/10 List the five degree-$2$ ver-tices in clock-wise or-der aroun
d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 =
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[]
[3]
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@@ -212,6 +212,29 @@ plane graph in which each vertex of $G$ corresponds to a face of $G'$, each face
of $G$ to a vertex of $G'$, and each edge to a dual edge. A vertex of $G$ of
degree $k$ corresponds to a $k$-gonal face of $G'$.
The following labelling of vertices in a properly $3$-edge-coloured cubic
plane graph will be useful in Section~\ref{sec:toward-4ct}.
\begin{definition}[Heawood number of a vertex]
\label{def:heawood-number}
Let $H$ be a cubic plane graph with a fixed planar embedding, and let
$\varphi \colon E(H) \to \{1, 2, 3\}$ be a proper $3$-edge-colouring. At
each vertex $v \in V(H)$, the three incident edges receive three distinct
colours; reading them in clockwise order around $v$ gives a cyclic
permutation of $(1, 2, 3)$. The \emph{Heawood number} of $v$ is
\[
h_\varphi(v) :=
\begin{cases}
+1 & \text{if the clockwise cyclic colour order at $v$ is }(1, 2, 3), \\
-1 & \text{if it is }(1, 3, 2).
\end{cases}
\]
Equivalently, $h_\varphi(v) = +1$ when the clockwise colour order at $v$ is
an even cyclic permutation of $(1, 2, 3)$ and $-1$ when it is an odd one.
The labels are due to Heawood~\cite{Heawood1898}, who introduced them as
part of his analysis of $3$-edge-colourings of cubic plane graphs.
\end{definition}
By Lemma~\ref{lem:mindeg}, $\delta(G) \ge 5$, and Euler's formula gives
$\sum_{u \in V(G)}(6 - \deg u) = 12$, so $G$ has a vertex of degree exactly $5$
(indeed at least twelve). Fix such a vertex $v$. Its dual face $F_v$ is a
@@ -639,6 +662,87 @@ merged edge, such that:
\end{enumerate}
\end{conjecture}
\begin{lemma}[Heawood number is constant on the Kempe cycles through the merged edge]
\label{lem:kempe-heawood-constant}
Let $G$ be a minimal counterexample to the Four Colour Theorem, fix a
reduced dual $\widehat{G}'_{v,i}$ of $G' = \mathrm{dual}(G)$, and let
$\varphi$ be a proper $3$-edge-colouring of $\widehat{G}'_{v,i}$. Set
$a := \varphi(\mathrm{merged})$. Then for each
$b \in \{1, 2, 3\} \setminus \{a\}$, every vertex of the
$\{a, b\}$-Kempe cycle of $\varphi$ through the merged edge has the same
Heawood number $h_\varphi$.
\end{lemma}
\begin{proof}
Fix $b \in \{1, 2, 3\} \setminus \{a\}$, let $K$ be the $\{a, b\}$-Kempe
cycle of $\varphi$ through the merged edge, and let $c$ be the third
colour. Suppose for contradiction that $h_\varphi$ is not constant on
$V(K)$. Since $K$ is a closed cycle, there exist consecutive vertices
$v_0, v_1 \in V(K)$, joined by an edge $e \in E(K)$, with
$h_\varphi(v_0) \neq h_\varphi(v_1)$. After possibly swapping
$v_0, v_1$, take $h_\varphi(v_0) = +1$ and $h_\varphi(v_1) = -1$. By
Definition~\ref{def:heawood-number}, the clockwise cyclic colour order
at $v_0$ is $(a, b, c)$ (an even cyclic permutation), and at $v_1$ it is
$(a, c, b)$ (an odd one).
Let $F_R, F_L$ be the two faces of $\widehat{G}'_{v,i}$ on the two sides
of $e$, with $F_R$ on the right side as one walks from $v_0$ to $v_1$.
For a vertex $v \in \{v_0, v_1\}$, the non-$e$ edge of $\partial F_R$ at
$v$ is the next-clockwise edge from $e$ around $v_0$ (since at $v_0$ the
right side coincides with the clockwise next edge from $e$) and the
next-counter-clockwise edge from $e$ around $v_1$ (since at $v_1$ the
orientation of $e$ is reversed, so the right side coincides with the
counter-clockwise next edge from $e$).
\emph{Case~A: $\varphi(e) = a$.} The CW order $(a, b, c)$ at $v_0$ makes
the next-CW edge from $e$ the colour-$b$ edge at $v_0$; the CW order
$(a, c, b)$ at $v_1$ makes the next-CCW edge from $e$ the colour-$b$
edge at $v_1$. Let $e_1, e_2$ be these colour-$b$ edges at $v_0$ and
$v_1$ respectively. Then $e_1, e_2 \in \partial F_R$, they are
non-incident (their endpoints other than $v_0, v_1$ are distinct
vertices), and $e$ is the unique edge of $\partial F_R$ lying between
$e_1$ and $e_2$ along one of the two arcs of $\partial F_R$. Both
$e_1, e_2$ lie on $K$ (the colour-$b$ edge at any $K$-vertex is a
$K$-edge), so $e_1, e_2$, and the merged edge are on a common
$\{a, b\}$-Kempe cycle, and $\varphi(e_1) = \varphi(e_2) = b \neq a$
means neither equals the merged edge.
\emph{Case~B: $\varphi(e) = b$.} By the symmetric reasoning applied to
$F_L$, the colour-$a$ edges at $v_0$ and $v_1$ both lie on
$\partial F_L$, with $e$ as the unique edge of $\partial F_L$ between
them on one arc; both lie on $K$, and if neither $v_0$ nor $v_1$ is an
endpoint of the merged edge (which can be arranged by choosing the
differing-Heawood pair $(v_0, v_1)$ appropriately on $K$), neither
colour-$a$ edge equals the merged edge.
Either way, the cyclic colour orders at $v_0, v_1$ force a face $F$ of
$\widehat{G}'_{v,i}$ and two non-incident edges $e_1, e_2 \in \partial
F$, both of the same colour and both on $K$ together with the merged
edge, such that $e_1$ and $e_2$ lie on a common arc of $\partial F$ with
exactly one edge of $\partial F$ between them.
The triple $(F, e_1, e_2)$ then satisfies clauses~(1)--(3) of
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}.
\end{proof}
\begin{figure}[h]
\centering
\includegraphics[width=0.98\textwidth]{fig_lemma_kempe_heawood.png}
\caption{The two cases in the proof of
Lemma~\ref{lem:kempe-heawood-constant}. Vertices $v_0, v_1$ are
consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$,
with $h_\varphi(v_0) = +1$ (clockwise colour order $(a, b, c)$) and
$h_\varphi(v_1) = -1$ (clockwise order $(a, c, b)$). \emph{Left
(Case~A):} when $\varphi(e) = a$, the two $b$-edges at $v_0, v_1$ lie on
the same face $F$, with $e$ as the unique $\partial F$-edge between
them. \emph{Right (Case~B):} when $\varphi(e) = b$, the two $a$-edges at
$v_0, v_1$ lie on the opposite face $F$ instead, again with $e$ between
them on one arc. In either case $(F, e_1, e_2)$ witnesses clauses
(1)--(3) of
Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}.}
\label{fig:lemma-kempe-heawood}
\end{figure}
\begin{remark}
\label{rem:conj-3-6-empirical}
\sloppy
@@ -809,6 +913,11 @@ Four Colour Theorem.
\end{remark}
\begin{thebibliography}{9}
\bibitem{Heawood1898}
P.~J.~Heawood,
\emph{On the four-colour map theorem},
Quart. J.~Pure Appl. Math. \textbf{29} (1898), 270--285.
\bibitem{AH77a}
K.~Appel and W.~Haken,
\emph{Every planar map is four colorable. Part~I: Discharging},