From c6d4eb06fb56903bc4a8cf764ac5ea59c3d1f350 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 19:48:55 -0400 Subject: [PATCH] coloring_nested_tire_graphs: remove complete tire dual definition and Tait correspondence The complete tire dual D*(T) and Tait correspondence material was giving exact counts but only via Tait colorings with XOR-to-0 conditions at high-degree vertices -- not proper edge colorings, which was the original goal. Removed: - Definition 1.13 (Complete tire dual) and its figure. - Proposition 1.14 (Tait correspondence on complete tire dual). - Remark 1.15 (Tait construction). - Remark 1.16 (octahedron verification). - Tait1880 bibitem (no longer cited; the body-text mention of Tait's theorem in the introduction remains). - experiments/draw_complete_tire_dual.py + its PNG. - experiments/draw_tait_coloring_example.py + its PNG. - fig_complete_tire_dual.png from the paper root. The partial tire dual definition (Def 1.7), structure proposition (Prop 1.8), figure, and associated scripts (tire_graph.py, draw_partial_tire_dual.py) are retained. Paper shrinks from 9 to 7 pages. Co-Authored-By: Claude Opus 4.7 --- .../complete_tire_dual_example.png | Bin 154249 -> 0 bytes .../experiments/draw_complete_tire_dual.py | 199 ------------------ .../fig_complete_tire_dual.png | Bin 154249 -> 0 bytes papers/coloring_nested_tire_graphs/paper.aux | 18 +- papers/coloring_nested_tire_graphs/paper.log | 63 +++--- papers/coloring_nested_tire_graphs/paper.pdf | Bin 748601 -> 616266 bytes papers/coloring_nested_tire_graphs/paper.tex | 90 -------- 7 files changed, 33 insertions(+), 337 deletions(-) delete mode 100644 papers/coloring_nested_tire_graphs/experiments/complete_tire_dual_example.png delete mode 100644 papers/coloring_nested_tire_graphs/experiments/draw_complete_tire_dual.py delete mode 100644 papers/coloring_nested_tire_graphs/fig_complete_tire_dual.png diff --git a/papers/coloring_nested_tire_graphs/experiments/complete_tire_dual_example.png b/papers/coloring_nested_tire_graphs/experiments/complete_tire_dual_example.png deleted file mode 100644 index 4aa8af4ec191925fd300b60be04cfecfc20f9729..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 154249 zcmd?R2T)U6{69#y(riemcIgU4l#Ub|2uPETA`p5Fy`zZIEEMTYdI>f3W&sJ(2?0V= zq=wK#gi!WeeeeDLJG(oxGdnx8Gka&=d+`Z3_nv#s_k7y-zS4Mjhw&)vC>0eIN*hl2JqdG;Nd>YAex>p}vZL0bF)9}p8ukJzRrH|K?**vbk z`1!~zV>zx)VrbrH&ta=WD&f1T3>@&14ObKd)jMjw2pa`?^He=b^Ffet(`c_mC|r~TL0 zr>S`agoV*nwm281KR1XtfA~BVRbZGe39EPG**6vnu6f@gHa1pAE=}62oRQxhU=^<$MvVryFOh1yj(lSEwwYBbu0Am zx3fMwPu11Ch%F>GQI;zdnQnc-iuQDOqrZPwB@7iOQASccCalBRM9{NKIY|!XXM_xD zrbBZ4j4DM&uMiEjWM}76eM$_Y4uph+x+LA^&G7csDJlLt3tK z(gOBtNh_0+Lnu?nV{F2e(4U8>f@v7}`1l-B=;`ToQEP7`>cmUYgzrZEfx0E{RDyyj?}4vDZxOR^J_-S~GL!#JWqnk|ZQHEhX-~5lge=>KE*>bkuW%BwmZePwzhUCSDgp%+|ES=7tOpa>9>`a8_ZB9E>GP4@sg)V zIFy-ZnTGXfnC$PVkn8rFoOVe`F#2fm;d*eQk@xDd)*G{3uFX7~K4x1>BU|g8B6zR4 z6kKmo!ZmT;dWZ44(23)`c56EkbYRj9++m?cTo%76?!AeSFe^%^#7NfcZQ+TOm1d7t zT8k^YTX@H<)z8d$S<2Eh4U@XuUZv3E()0`oE_p6%D&Nsc>*c0kIzzNwejQ)nHLGq; zpXo@6rOJVat_ep9b&JcVqolgIJ*R>KJ1O9C+BNTORZK_AO*HMUKd>Ds&*q5n-d$@a zCfb&i;T`IQrU;kS5})z-t_F3&YF0lTE0oza&TuN-~Y{IPl)Un~7(u zMdCa;pB=xyAXq{G{7P zF05B#%qb+PZ!AZRLjsZKy_1{jZS$>ec0cm;>5m$^&qcXU4ffh0EbiEiR^~6l@7L-j zoa&O-bybg5)2zJ|%XY*=#^!{i<@d9M(JEUk+Qg}q9&I-w8m@fgja8c85U3{CB=QO# zT}vD(M#^mso9lvuxQ$uwMdwSr$kPrF2pKkOJaXPx&pvY`PGj%nZgf2+ErLU$+k0zi z0S)i9ecGrV^gR*YX`FO4%o^<-%`Iaz!i{QBzd9P_1j`Av=`IR_HHwI@<^^#OO>Jxn zs_gMY?jsg#67^CONNm9NOnd>Qik`KB_h-67$e5qSrP!F5=e|u0+O~xJAB?*B9zltl zp@imxJrpG|Cgz#qOD0AFJU7|pHKpMDYo|md95RVOs1Pza^>|BEr=)I?|AF?7bgz$I zyFrkk$W4>9n0&D(pXfZg_t&BW1}QtIhaa8!GSwPl9Khsfx#*EZV>g%V`miPSV%cuW zS>npU<7E8KZ$C|(-=^FU=6k>Y0y)b6;A-cC)+@2m5#p3=okCrQkwse#4*Kd%Bch_T z+y$^dfNDsC=wIL15YT$`h@a%n1Gmvmi6?($Gu1|e+Z7TSCXU*o7=MorJe2B^Y3h-s zXv>nvTx0X|^Qlece1ETEK!wyRF|00e`4YxToi+eAG>lcMSI!Vg8a$KOj*brM_`U+2 zLJX>TUvf&+pol_2fq@@?*rp_MqH zKusKfVzK5oH~A%xe@&Y+IY3Q%t79_Bu}ME6HI>_GBq1RocBP3@f*1Bf`EBI-^_h=W zYupx6#B1Q4m&a;dXGEHJPmrn}C($z9Y!S)=-}j_>7F6}T(I$2=OYI>YT#4~`d0R9O zPagrS93OIB5bdGRy~G#C*mUhFpm>_Huh^HMdeEM?HhUGv>wGrnHHmXzBaNMznIWOM zn%etJ6%-U^4)!JwYTHSv{=_OAKf%!2+8Rsx-4n21u{bwTTS#=xEHUxlo=L~Wc?^`j zYG3*Ccz-{7c5JA9CA@Aq>%f1}*W%eTI-^kk{Sm7)<;{cuQ26xxq6m%b-``z&!2kZ; zyD?WD@A-@xe-(gf(zkg}QX_sgf%;xB;^ni_=#dhaJ^Cw=y5`Ae28m-@YINeNdv7L&3)g*%wdz|kvCoAmPlE4rV3?cjN+O0`h;mYUb|G7( zq~M^0y~h7w~7GUWy4a6u<{*n9sz?8hrvQB|TWF-Ow<2Cif0I>08su%(qo zvPyOi?JPMQ-`!mFxa|D1ws+;n+>USV>!@fq$o^TQ-^JZd(aQT;4`oQ;m$Ktc3I@tm zqPdB2SqX3C_J5m4wjVooY+TjQF)rmGLf$bn)gf^khPXUl<2*e!jY;h#7NVL8){;jm ztm;SVyeG1vWeujCCrQV-Br9Dyb^)`BAozsI;5TPrr+?iihCv?Y1SgmzQu}b9bxdSIBpzw|~9Y=`^Tyu^2`T zcx{+?%%ym!RCZs8mhNwKAdh=QQ*PjJR!J*MKM;=lxN_O*#`bPWf}li_)UYk7c7L%D zrQ7x(ePl8Gj8MUtJKo;5Yxq@r*&5tq*{J#;@#{)6k_mqbv4BLkBYo`BndXz` zB#K<3t07Ob$kVduCQ3d&=n(f~{YQ@;P2WN_tl!7F%I>ay4wK!e#yd!nhp)GDo^;~j z2S@dYB?meAq`LI=lE0|KIuIB0YRlzUmLpzW z`n0tN`;_tH_5lEIM}OQxn}(8g|KzkeRxQTX-i7hTb||7PDEJuEG% zc(y*k{wLUfKaJzQ_n1}n!|?-2^uIqH6M5da8)~2@Xl+@zs3S2byWs7&(nucj>q;Lg z-_7U#ZMwhCW{aqkjeRji;z4}N!L-UsNxCmR&b40^H4w*-B<;bXb1+TfhK8RO{8&5x zl(xrfU0?c^+>7-{W3gX#wY7hh^Kl34I+H1iX#|^VL?4r%_bR%uca+446cS<1X8Vpe z*O%9_cPbC!jotM_>_QwH56slMKC^a#nSQVz{cCIpx6^9Pd8zTl+^EcM;!lF+$C6D74=lioAX@&PuCi8u9U zU%drkbPmB|B=ut4*=uK4yzUd2RYC$X;(o-cYOsgjW%%kQC|-Jwzl^0ox~=rd=YrT{ zueAM21a!3}@6O1=T+)0|dX9mzEC=Q6Yn)WRvk6v9-!4SES!9)$mlwcGbrVkY1sX^v z@!t{sy}+nsJyj$e68DqQeP(u6MJM#$d9E7LP&VOeqNrw{ooj;}UOQOnvNeP-!Z(DogqOb@4L+un4jZPc}B#< z2pOC{52X zwUyiY47)=1wxUZ^GvraAbHv9U$>2WomZ8EIH<@)Oy2jDf&kz0e-9eEX{b8Dv6`8H! 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13c8226..0000000 --- a/papers/coloring_nested_tire_graphs/experiments/draw_complete_tire_dual.py +++ /dev/null @@ -1,199 +0,0 @@ -"""Draw the complete tire dual D*(T) on top of a small tire graph. - -For a spoke-only tire with 2-connected O = C_k (no chords), D*(T) is -obtained from D(T) by merging: - - all B_out-leaves into a single outer-face vertex v_out (degree n); - - the B_in-leaves into a single inner-face vertex v_in (degree k), - because O has exactly one bounded interior face. - -Equivalently, D*(T) is the planar dual of T. -""" -import math -import os -import sys - -HERE = os.path.dirname(os.path.abspath(__file__)) -sys.path.insert(0, HERE) - -import matplotlib.pyplot as plt -import matplotlib.patches as patches - -from tire_graph import random_tire, planar_positions - - -def draw_complete_tire_dual(tire, filename): - m, k = tire['m'], tire['k'] - outer, inner = tire['outer'], tire['inner'] - triangles = tire['triangles'] - - R_out, R_in = 1.0, 0.45 - pos = planar_positions(tire, R_out=R_out, R_in=R_in) - - fig, ax = plt.subplots(figsize=(9, 9)) - - # === Draw underlying tire graph faintly === - outer_set = set(outer) - inner_set = set(inner) - for u, v in tire['edges']: - x1, y1 = pos[u]; x2, y2 = pos[v] - if u in outer_set and v in outer_set: - color = '#a8c9e8'; lw = 2.0 - elif u in inner_set and v in inner_set: - color = '#e8a8a8'; lw = 2.0 - else: - color = '#cccccc'; lw = 1.0 - ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1) - for v in outer: - x, y = pos[v] - ax.plot(x, y, 'o', color='#a8c9e8', markersize=14, zorder=2) - ax.annotate(str(v), (x, y), color='white', ha='center', va='center', - fontsize=8, fontweight='bold', zorder=3) - for v in inner: - x, y = pos[v] - ax.plot(x, y, 'o', color='#e8a8a8', markersize=12, zorder=2) - ax.annotate(str(v), (x, y), color='white', ha='center', va='center', - fontsize=7, fontweight='bold', zorder=3) - - n_tri = len(triangles) - centroids = [] - for tri in triangles: - v1, v2, v3 = tri - cx = (pos[v1][0] + pos[v2][0] + pos[v3][0]) / 3 - cy = (pos[v1][1] + pos[v2][1] + pos[v3][1]) / 3 - centroids.append((cx, cy)) - - # Identify which triangles are incident to a B_out edge (O-move type) - # vs incident to a B_in edge (I-move type). - from collections import defaultdict - bout_edges = {frozenset({outer[i], outer[(i+1) % m]}) for i in range(m)} - bin_edges = {frozenset({inner[j], inner[(j+1) % k]}) for j in range(k)} - - bout_incident = [] # triangles incident to a B_out edge - bin_incident = [] # triangles incident to a B_in edge - for t_idx, tri in enumerate(triangles): - tri_edges = {frozenset({tri[0], tri[1]}), - frozenset({tri[1], tri[2]}), - frozenset({tri[0], tri[2]})} - if tri_edges & bout_edges: - bout_incident.append(t_idx) - if tri_edges & bin_edges: - bin_incident.append(t_idx) - - print(f"O-move triangles (B_out incident): {bout_incident}") - print(f"I-move triangles (B_in incident): {bin_incident}") - - # Place v_out outside the outer cycle (far to the upper right) - # and v_in at the origin (centroid of inner cycle). - v_out_pos = (1.55, 0.0) # well outside the outer cycle - v_in_pos = (0.0, 0.0) # center of inner cycle - - # Identify dual cycle (interior dual edges) - edge_to_tris = defaultdict(list) - for t_idx, tri in enumerate(triangles): - e1 = frozenset({tri[0], tri[1]}) - e2 = frozenset({tri[1], tri[2]}) - e3 = frozenset({tri[0], tri[2]}) - for e in (e1, e2, e3): - edge_to_tris[e].append(t_idx) - - tri_adj = defaultdict(set) - for e, ts in edge_to_tris.items(): - if len(ts) == 2: - t1, t2 = ts - tri_adj[t1].add(t2) - tri_adj[t2].add(t1) - - # === Draw complete tire dual === - DUAL_COLOR = '#7d3c98' # purple - OUT_COLOR = '#1f77b4' # blue (for v_out edges) - IN_COLOR = '#d62728' # red (for v_in edges) - - # Interior dual edges (d_f to d_f) - for t1 in range(n_tri): - for t2 in tri_adj[t1]: - if t2 <= t1: - continue - x1, y1 = centroids[t1] - x2, y2 = centroids[t2] - ax.plot([x1, x2], [y1, y2], color=DUAL_COLOR, linewidth=2.0, - zorder=4) - - # v_out edges (to all O-move d_f's) - for t_idx in bout_incident: - cx, cy = centroids[t_idx] - ax.plot([cx, v_out_pos[0]], [cy, v_out_pos[1]], - color=OUT_COLOR, linewidth=1.6, linestyle='--', zorder=4) - - # v_in edges (to all I-move d_f's) - for t_idx in bin_incident: - cx, cy = centroids[t_idx] - ax.plot([cx, v_in_pos[0]], [cy, v_in_pos[1]], - color=IN_COLOR, linewidth=1.6, linestyle='--', zorder=4) - - # Interior dual vertices d_f - for t_idx, (cx, cy) in enumerate(centroids): - ax.plot(cx, cy, 's', color=DUAL_COLOR, markersize=14, zorder=5) - ax.annotate(f"$d_{{{t_idx}}}$", (cx, cy), color='white', - ha='center', va='center', fontsize=7, fontweight='bold', - zorder=6) - - # v_out (outer-face vertex) - ax.plot(v_out_pos[0], v_out_pos[1], 'h', color=OUT_COLOR, - markersize=20, zorder=5) - ax.annotate("$v_{\\mathrm{out}}$", v_out_pos, color='white', - ha='center', va='center', fontsize=9, fontweight='bold', - zorder=6) - - # v_in (inner-face vertex) - ax.plot(v_in_pos[0], v_in_pos[1], 'h', color=IN_COLOR, - markersize=18, zorder=5) - ax.annotate("$v_{\\mathrm{in}}$", v_in_pos, color='white', - ha='center', va='center', fontsize=8, fontweight='bold', - zorder=6) - - # Legend - legend_items = [ - plt.Line2D([], [], marker='o', color='w', markerfacecolor='#a8c9e8', - markersize=12, label='$B_{\\mathrm{out}}$ vertex (tire)'), - plt.Line2D([], [], marker='o', color='w', markerfacecolor='#e8a8a8', - markersize=11, label='$B_{\\mathrm{in}}$ vertex (tire)'), - plt.Line2D([], [], marker='s', color='w', markerfacecolor=DUAL_COLOR, - markersize=12, label='$d_f$ (annular face dual vertex)'), - plt.Line2D([], [], marker='h', color='w', markerfacecolor=OUT_COLOR, - markersize=14, - label='$v_{\\mathrm{out}}$ (outer-face vertex, deg ${%d}$)' % m), - plt.Line2D([], [], marker='h', color='w', markerfacecolor=IN_COLOR, - markersize=13, - label='$v_{\\mathrm{in}}$ (inner-face vertex, deg ${%d}$)' % k), - plt.Line2D([], [], color=DUAL_COLOR, linewidth=2.0, - label='interior dual edge ($d_f$–$d_{f^\\prime}$)'), - plt.Line2D([], [], color=OUT_COLOR, linewidth=1.6, linestyle='--', - label='dual edge to $v_{\\mathrm{out}}$'), - plt.Line2D([], [], color=IN_COLOR, linewidth=1.6, linestyle='--', - label='dual edge to $v_{\\mathrm{in}}$'), - ] - ax.legend(handles=legend_items, loc='upper left', - bbox_to_anchor=(1.02, 1.0), fontsize=9, frameon=False) - - ax.set_xlim(-1.35, 2.05) - ax.set_ylim(-1.35, 1.35) - ax.set_aspect('equal') - ax.axis('off') - ax.set_title( - f'Complete tire dual $D^{{*}}(T)$ (m={m}, k={k}, spoke-only)\n' - f'underlying tire faint; $|V| = {n_tri + 2}$, $|E| = {2*n_tri}$', - fontsize=12) - - plt.savefig(filename, dpi=160, bbox_inches='tight') - plt.close() - print(f"wrote {filename}") - - -def main(): - tire = random_tire(m=6, k=4, n_chords=0, seed=3) - out = os.path.join(HERE, 'complete_tire_dual_example.png') - draw_complete_tire_dual(tire, out) - - -if __name__ == '__main__': - main() diff --git 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b/papers/coloring_nested_tire_graphs/paper.aux @@ -15,21 +15,13 @@ \newlabel{lem:tire-component}{{1.10}{5}} \citation{bauerfeld-pds} \citation{bauerfeld-pds} -\newlabel{rem:tire-component-degenerate}{{1.11}{7}} -\newlabel{rem:tire-no-extra-hypotheses}{{1.12}{7}} -\newlabel{def:complete-tire-dual}{{1.13}{7}} -\citation{Tait1880} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The complete tire dual $D^{\ast }(T)$ (purple squares + face hexagons) drawn on top of the same $m = 6$, $k = 4$ spoke-only tire graph as Figure\nonbreakingspace 3\hbox {}. The ten annular-face vertices $d_f$ form a $10$-cycle (solid purple); the $n$ outer leaves of $D(T)$ have been merged into a single outer-face vertex $v_{\mathrm {out}}$ (blue hexagon, drawn outside the tire) of degree $n = 6$, and the $m$ inner leaves into a single inner-face vertex $v_{\mathrm {in}}$ (red hexagon, at the centre of the inner cycle) of degree $m = 4$. 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