From 037d987c7de75dc48f32a2d48ffafa5488f73055 Mon Sep 17 00:00:00 2001 From: didericis Date: Sun, 24 May 2026 22:31:10 -0400 Subject: [PATCH] face_monochromatic_pairs: reframe Lemma 5.2 as a non-existence result 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 --- .../experiments/draw_lemma_kempe_heawood.py | 168 ++++++++---------- .../fig_lemma_kempe_heawood.png | Bin 72041 -> 63270 bytes papers/face_monochromatic_pairs/paper.aux | 4 +- papers/face_monochromatic_pairs/paper.log | 18 +- papers/face_monochromatic_pairs/paper.pdf | Bin 1081744 -> 1077738 bytes papers/face_monochromatic_pairs/paper.tex | 117 ++++++------ 6 files changed, 148 insertions(+), 159 deletions(-) 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: $\\varphi(e) = b$. 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[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). 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