diff --git a/papers/face_monochromatic_pairs/experiments/draw_lemma_kempe_heawood.py b/papers/face_monochromatic_pairs/experiments/draw_lemma_kempe_heawood.py index bb9ece1..6e3ae21 100644 --- a/papers/face_monochromatic_pairs/experiments/draw_lemma_kempe_heawood.py +++ b/papers/face_monochromatic_pairs/experiments/draw_lemma_kempe_heawood.py @@ -1,17 +1,17 @@ -"""Two-panel illustration of the proof of Lemma 5.2 -(Heawood constant on Kempe cycles through merged). +"""Two-panel illustration of the proof of Lemma 5.2 (a Heawood-constant +Kempe cycle does not admit the clause-(3) arc of Conjecture 5.1). 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)). +cycle K, joined by an edge e, with h(v_0) = h(v_1) = +1: i.e., both +have the same clockwise colour order (a, b, c). The would-be witness +edges (b-edges in Case A, a-edges in Case B) lie on opposite sides of +e, so no face of the graph contains both of them. -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). +Left panel (Case A): phi(e) = a. The b-edges at v_0, v_1 are on + opposite sides of e (one south, one north). -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). +Right panel (Case B): phi(e) = b. The a-edges at v_0, v_1 are on + opposite sides of e. Produces fig_lemma_kempe_heawood.png. """ @@ -25,12 +25,12 @@ 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' +FACE_FILL_R = '#fef3c7' # F_R shading (south) +FACE_FILL_L = '#dbeafe' # F_L shading (north) V0 = (-1.6, 0.0) V1 = ( 1.6, 0.0) @@ -62,7 +62,7 @@ def label_text(ax, p, text, color=DARK, fontsize=12, dx=0, dy=0, edgecolor='none', alpha=0.85)) -def label_edge_midpoint(ax, p, q, text, color, fontsize=11, offset=(0, 0)): +def label_edge_midpoint(ax, p, q, text, color, fontsize=10, 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', @@ -71,132 +71,122 @@ def label_edge_midpoint(ax, p, q, text, color, fontsize=11, offset=(0, 0)): edgecolor='none', alpha=0.9)) -def shade_face(ax, pts, color=FACE_FILL, alpha=0.7): +def shade_face(ax, pts, color, alpha=0.55): 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 + # Same Heawood: v_0 and v_1 both have CW order (a, b, c) with e = a. + # v_0: e at 0 deg (east), b at 300 deg (south), c at 60 deg (north). + # v_1: e at 180 deg (west), b at 90 deg (north), c at 270 deg (south). + # The b-edges land on opposite sides of e (south of v_0, north of v_1). + b0 = edge_at(V0, 300) # south of v_0 + c0 = edge_at(V0, 60) # north of v_0 + b1 = edge_at(V1, 90) # north of v_1 + c1 = edge_at(V1, 270) # south of v_1 - # 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') + # Shade both F_R (south) and F_L (north) lightly. + shade_face(ax, [V0, V1, c1, (c1[0] + 0.3, c1[1] - 0.6), + (b0[0] - 0.3, b0[1] - 0.6), b0], color=FACE_FILL_R) + shade_face(ax, [V0, c0, (c0[0] - 0.3, c0[1] + 0.6), + (b1[0] + 0.3, b1[1] + 0.6), b1, V1], color=FACE_FILL_L) + label_text(ax, ((V0[0] + V1[0]) / 2, -1.7), '$F_R$', color=DARK, + fontsize=11, weight='bold') + label_text(ax, ((V0[0] + V1[0]) / 2, 1.7), '$F_L$', color=DARK, + fontsize=11, 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 + draw_edge(ax, V0, V1, COL_A) + draw_edge(ax, V0, b0, COL_B) + draw_edge(ax, V0, c0, COL_C) + draw_edge(ax, V1, b1, COL_B) + draw_edge(ax, V1, c1, COL_C) - # Vertices - draw_vertex(ax, V0, DARK) - draw_vertex(ax, V1, DARK) + 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) + draw_stub(ax, b1); 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), '$h_\\varphi\\!=\\!-1$', + 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)) + offset=(0, 0.18)) + label_edge_midpoint(ax, V0, b0, '$b$', color=COL_B, + offset=(-0.15, 0)) label_edge_midpoint(ax, V0, c0, '$c$', color=COL_C, - offset=(0.05, 0)) + offset=(-0.15, 0)) + label_edge_midpoint(ax, V1, b1, '$b$', color=COL_B, + offset=(0.15, 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)) + offset=(0.15, 0)) - 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') + ax.set_title('Case A: $\\varphi(e) = a$. The two $b$-edges are on\n' + 'opposite sides of $e$ -- no common face', + fontsize=11, color=DARK, pad=8, 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 + # Same Heawood: v_0 and v_1 both have CW order (a, b, c) with e = b. + # v_0: a at 60 deg (north), e (b) at 0 deg (east), c at 300 deg (south). + # v_1: a at 270 deg (south), e (b) at 180 deg (west), c at 90 deg (north). + # The a-edges land on opposite sides of e (north of v_0, south of v_1). + a0 = edge_at(V0, 60) # north of v_0 + c0 = edge_at(V0, 300) # south of v_0 + a1 = edge_at(V1, 270) # south of v_1 + c1 = edge_at(V1, 90) # north of v_1 - # 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') + shade_face(ax, [V0, V1, a1, (a1[0] + 0.3, a1[1] - 0.6), + (c0[0] - 0.3, c0[1] - 0.6), c0], color=FACE_FILL_R) + shade_face(ax, [V0, a0, (a0[0] - 0.3, a0[1] + 0.6), + (c1[0] + 0.3, c1[1] + 0.6), c1, V1], color=FACE_FILL_L) + label_text(ax, ((V0[0] + V1[0]) / 2, -1.7), '$F_R$', color=DARK, + fontsize=11, weight='bold') + label_text(ax, ((V0[0] + V1[0]) / 2, 1.7), '$F_L$', color=DARK, + fontsize=11, weight='bold') - # Edges - draw_edge(ax, V0, V1, e_color) + draw_edge(ax, V0, V1, COL_B) 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_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), '$h_\\varphi\\!=\\!-1$', + 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, a0, '$a$', color=COL_A, + offset=(-0.15, 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)) + offset=(-0.15, 0)) + label_edge_midpoint(ax, V1, a1, '$a$', color=COL_A, + offset=(0.15, 0)) label_edge_midpoint(ax, V1, c1, '$c$', color=COL_C, - offset=(-0.05, 0)) + offset=(0.15, 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') + ax.set_title('Case B: $\\varphi(e) = b$. The two $a$-edges are on\n' + 'opposite sides of $e$ -- no common face', + fontsize=11, color=DARK, pad=8, fontweight='bold') def main(): - plt.rcParams['text.usetex'] = False # keep matplotlib defaults - fig, axes = plt.subplots(1, 2, figsize=(13, 5.5)) + fig, axes = plt.subplots(1, 2, figsize=(13, 5.8)) for ax in axes: ax.set_xlim(-3.5, 3.5) - ax.set_ylim(-2.4, 2.4) + ax.set_ylim(-2.5, 2.5) ax.set_aspect('equal') ax.axis('off') panel_case_A(axes[0]) diff --git a/papers/face_monochromatic_pairs/fig_lemma_kempe_heawood.png b/papers/face_monochromatic_pairs/fig_lemma_kempe_heawood.png index 015af9a..deb7f87 100644 Binary files a/papers/face_monochromatic_pairs/fig_lemma_kempe_heawood.png and b/papers/face_monochromatic_pairs/fig_lemma_kempe_heawood.png differ diff --git a/papers/face_monochromatic_pairs/paper.aux b/papers/face_monochromatic_pairs/paper.aux index 9f5245e..76442fb 100644 --- a/papers/face_monochromatic_pairs/paper.aux +++ b/papers/face_monochromatic_pairs/paper.aux @@ -38,9 +38,9 @@ \newlabel{conj:face-monochromatic-pair-on-merged-kempe-cycle}{{5.1}{10}} \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 } +\@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 the lemma's hypothesis $h_\varphi (v_0) = h_\varphi (v_1) = +1$ --- so both vertices share the clockwise colour order $(a, b, c)$. \emph {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the colour-$b$ edge at $v_0$ lies south of $e$ (on $\partial F_R$) and the colour-$b$ edge at $v_1$ lies north of $e$ (on $\partial F_L$); the two would-be witness edges are on opposite faces, so no face of $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,i}$ contains both. \emph {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the colour-$a$ edges at $v_0, v_1$ are likewise on opposite sides of $e$. In either case the clause-$(3)$ arc of Conjecture\nonbreakingspace 5.1\hbox {} cannot be realised at $e$.}}{12}{}\protected@file@percent } \newlabel{fig:lemma-kempe-heawood}{{5}{12}} -\newlabel{conj:face-monochromatic-pair-strengthened}{{5.4}{13}} +\newlabel{conj:face-monochromatic-pair-strengthened}{{5.4}{12}} \newlabel{rem:conj-3-8-empirical}{{5.5}{13}} \newlabel{rem:implication-4ct}{{5.6}{13}} \bibcite{Heawood1898}{1} diff --git a/papers/face_monochromatic_pairs/paper.log b/papers/face_monochromatic_pairs/paper.log index 02ee769..1a7ea8d 100644 --- a/papers/face_monochromatic_pairs/paper.log +++ b/papers/face_monochromatic_pairs/paper.log @@ -1,4 +1,4 @@ -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 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 24 MAY 2026 22:13 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -262,23 +262,23 @@ Package pdftex.def Info: fig_thm_cubic_contraction_4face.png used on input lin e 620. (pdftex.def) Requested size: 352.79846pt x 160.78339pt. [10 <./fig_thm_cubic_contraction_4face.png>] - + File: fig_lemma_kempe_heawood.png Graphic file (type png) -Package pdftex.def Info: fig_lemma_kempe_heawood.png used on input line 730. -(pdftex.def) Requested size: 352.79846pt x 129.2451pt. +Package pdftex.def Info: fig_lemma_kempe_heawood.png used on input line 727. +(pdftex.def) Requested size: 352.79846pt x 138.98488pt. LaTeX Warning: `h' float specifier changed to `ht'. [11] [12 <./fig_lemma_kempe_heawood.png>] -Underfull \hbox (badness 1648) in paragraph at lines 825--831 +Underfull \hbox (badness 1648) in paragraph at lines 824--830 \OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.5\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T he strength-ened con-jec-ture was tested on the same chord- [] -Underfull \hbox (badness 1014) in paragraph at lines 825--831 +Underfull \hbox (badness 1014) in paragraph at lines 824--830 \OT1/cmr/m/n/10 apex+Kempe colour-ings as Re-mark 5.3[]; for each colour-ing we sought any [] @@ -287,11 +287,11 @@ Underfull \hbox (badness 1014) in paragraph at lines 825--831 Here is how much of TeX's memory you used: 3108 strings out of 478268 44593 string characters out of 5846347 - 348397 words of memory out of 5000000 + 349397 words of memory out of 5000000 21140 multiletter control sequences out of 15000+600000 478386 words of font info for 63 fonts, out of 8000000 for 9000 1302 hyphenation exceptions out of 8191 - 69i,12n,76p,875b,344s stack positions out of 10000i,1000n,20000p,200000b,200000s + 69i,12n,76p,1047b,344s stack positions out of 10000i,1000n,20000p,200000b,200000s -Output written on paper.pdf (14 pages, 1081744 bytes). +Output written on paper.pdf (14 pages, 1077738 bytes). PDF statistics: 192 PDF objects out of 1000 (max. 8388607) 106 compressed objects within 2 object streams diff --git a/papers/face_monochromatic_pairs/paper.pdf b/papers/face_monochromatic_pairs/paper.pdf index 8a0b26b..40e1d2e 100644 Binary files a/papers/face_monochromatic_pairs/paper.pdf and b/papers/face_monochromatic_pairs/paper.pdf differ diff --git a/papers/face_monochromatic_pairs/paper.tex b/papers/face_monochromatic_pairs/paper.tex index 77a7d7b..f0be990 100644 --- a/papers/face_monochromatic_pairs/paper.tex +++ b/papers/face_monochromatic_pairs/paper.tex @@ -662,67 +662,64 @@ merged edge, such that: \end{enumerate} \end{conjecture} -\begin{lemma}[Heawood number is constant on the Kempe cycles through the merged edge] +\begin{lemma}[A Heawood-constant Kempe cycle does not admit the clause-(3) arc] \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 $h_\varphi$. +$a := \varphi(\mathrm{merged})$ and let $K$ be the $\{a, b\}$-Kempe +cycle of $\varphi$ through the merged edge for some +$b \in \{1, 2, 3\} \setminus \{a\}$. If $h_\varphi$ is constant on +$V(K)$, then no edge $e \in E(K)$ admits a face $F$ of +$\widehat{G}'_{v,i}$ and two non-incident edges +$e_1, e_2 \in \partial F$ such that +$\varphi(e_1) = \varphi(e_2)$ and $e$ is the unique edge of +$\partial F$ between $e_1$ and $e_2$ along one of the two arcs of +$\partial F$ --- that is, no edge of $K$ admits the clause-(3) arc of +Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} +with $e_1, e_2$ at its two endpoints. \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 $c$ be the third colour. Fix any edge $e \in E(K)$ joining +$v_0, v_1 \in V(K)$. By hypothesis $h_\varphi(v_0) = h_\varphi(v_1)$; +after possibly relabelling we may take +$h_\varphi(v_0) = h_\varphi(v_1) = +1$, so by +Definition~\ref{def:heawood-number} the clockwise cyclic colour order +at $v_0$ and at $v_1$ is the same even cyclic class $(a, b, c)$. -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 $e$). +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~A: $\varphi(e) = a$.} In the CW order $(a, b, c)$ at $v_0$ +the next-CW edge from $e$ has colour $b$; in the same CW order +$(a, b, c)$ at $v_1$ the next-CCW edge from $e$ has colour $c$ (since +CCW-next from $a$ in cyclic order $(a, b, c)$ is $c$). Hence the +non-$e$ edge of $\partial F_R$ at $v_0$ has colour $b$, while the +non-$e$ edge of $\partial F_R$ at $v_1$ has colour $c$ --- these +differ. Symmetrically, the non-$e$ edges of $\partial F_L$ at $v_0$ +and $v_1$ have colours $c$ and $b$ respectively, again different. +Hence the colour-$b$ edges at $v_0$ and $v_1$ lie on opposite faces +of $e$, and the same for the colour-$c$ edges; no face of +$\widehat{G}'_{v,i}$ contains two same-coloured non-$e$ edges at +$\{v_0, v_1\}$. -\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. +\emph{Case~B: $\varphi(e) = b$.} By the analogous reasoning, the +non-$e$ edges of $\partial F_R$ at $v_0$ and $v_1$ have colours $c$ +and $a$ respectively, and those of $\partial F_L$ have colours $a$ +and $c$. The colour-$a$ edges at $v_0, v_1$ thus lie on opposite +faces of $e$, and so do the colour-$c$ edges. -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}. +In either case, no face $F$ of $\widehat{G}'_{v,i}$ has two same-coloured +non-$e$ edges at $\{v_0, v_1\}$ on $\partial F$, so the clause-(3) arc +(with $e$ as the unique $\partial F$-edge between $e_1$ and $e_2$ at the +endpoints of $e$) cannot be realised. \end{proof} \begin{figure}[h] @@ -731,15 +728,17 @@ Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle}. \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}.} +with the lemma's hypothesis $h_\varphi(v_0) = h_\varphi(v_1) = +1$ --- +so both vertices share the clockwise colour order $(a, b, c)$. +\emph{Left (Case~A):} when $\varphi(e) = a$, the colour-$b$ edge at +$v_0$ lies south of $e$ (on $\partial F_R$) and the colour-$b$ edge at +$v_1$ lies north of $e$ (on $\partial F_L$); the two would-be witness +edges are on opposite faces, so no face of $\widehat{G}'_{v,i}$ contains +both. \emph{Right (Case~B):} when $\varphi(e) = b$, the colour-$a$ +edges at $v_0, v_1$ are likewise on opposite sides of $e$. In either +case the clause-$(3)$ arc of +Conjecture~\ref{conj:face-monochromatic-pair-on-merged-kempe-cycle} +cannot be realised at $e$.} \label{fig:lemma-kempe-heawood} \end{figure}