didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

dual_decomposition: swap algorithm trace to n=14 + final-graph conjecture

Browse Source 
- Replace the dodecahedron trace at the end of section 3 with the n=14
  triangulation found by search_kempe_property.py: its H_1 admits a
  proper 3-edge-colouring satisfying both chord-apex and Kempe-cycle
  conditions (Lemmas 2.6, 2.7).
- experiments/draw_iterated_reduction_n14.py: rebuilds fig_alg_step{0,1,2}
  with Tutte barycentric layouts (outer face chosen to keep v_n in the
  interior); also runs the algorithm to completion, checking chord-apex +
  Kempe at each step (step 1 satisfies all; step 2 fails chord-apex;
  step 3 terminates).
- Add Conjecture 3.4: G is a minimal counterexample iff no proper
  3-edge-colouring of the final reduced graph H_{t*} has all (spike_t,
  merged_t) pairs in distinct colours.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-23 13:20:32 -04:00
parent c987259c14
commit 03dcd7c2fa
8 changed files with 630 additions and 59 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/dual_decomposition_minimal_counterexamples/paper.log
+78 -48
View File
@@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 23 MAY 2026 03:21
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 23 MAY 2026 13:19
entering extended mode
restricted \write18 enabled.
%&-line parsing enabled.
@@ -144,26 +144,26 @@ File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX)
(./paper.aux)
\openout1 = `paper.aux'.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 26.
LaTeX Font Info: ... okay on input line 26.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 26.
LaTeX Font Info: ... okay on input line 26.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 26.
LaTeX Font Info: ... okay on input line 26.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 26.
LaTeX Font Info: ... okay on input line 26.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 26.
LaTeX Font Info: ... okay on input line 26.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 26.
LaTeX Font Info: ... okay on input line 26.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 26.
LaTeX Font Info: ... okay on input line 26.
LaTeX Font Info: Trying to load font information for U+msa on input line 26.
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for OMS/cmsy/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for OT1/cmr/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for T1/cmr/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for TS1/cmr/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 27.
LaTeX Font Info: ... okay on input line 27.
LaTeX Font Info: Trying to load font information for U+msa on input line 27.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
)
LaTeX Font Info: Trying to load font information for U+msb on input line 26.
LaTeX Font Info: Trying to load font information for U+msb on input line 27.
(/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
@@ -192,7 +192,7 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Liv
e
))
[1{/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}]
Overfull \hbox (41.917pt too wide) in paragraph at lines 144--146
Overfull \hbox (41.917pt too wide) in paragraph at lines 145--147
[]\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 $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 =
(\OML/cmm/m/it/10 A[]; A[]; A[]; A[]; A[]\OT1/cmr/m/n/10 )$.
@@ -201,22 +201,22 @@ d $\OML/cmm/m/it/10 F$ \OT1/cmr/m/n/10 as $\OML/cmm/m/it/10 A \OT1/cmr/m/n/10 =
<fig_reduced_dual_step1.png, id=17, 517.79329pt x 499.08812pt>
File: fig_reduced_dual_step1.png Graphic file (type png)
<use fig_reduced_dual_step1.png>
Package pdftex.def Info: fig_reduced_dual_step1.png used on input line 164.
Package pdftex.def Info: fig_reduced_dual_step1.png used on input line 165.
(pdftex.def) Requested size: 172.79846pt x 166.55775pt.
<fig_reduced_dual_step2.png, id=19, 490.16064pt x 483.35876pt>
File: fig_reduced_dual_step2.png Graphic file (type png)
<use fig_reduced_dual_step2.png>
Package pdftex.def Info: fig_reduced_dual_step2.png used on input line 165.
Package pdftex.def Info: fig_reduced_dual_step2.png used on input line 166.
(pdftex.def) Requested size: 172.79846pt x 170.39505pt.
<fig_reduced_dual_step3.png, id=20, 490.16064pt x 483.35876pt>
File: fig_reduced_dual_step3.png Graphic file (type png)
<use fig_reduced_dual_step3.png>
Package pdftex.def Info: fig_reduced_dual_step3.png used on input line 166.
Package pdftex.def Info: fig_reduced_dual_step3.png used on input line 167.
(pdftex.def) Requested size: 172.79846pt x 170.39505pt.
<fig_reduced_dual_step4.png, id=21, 490.16064pt x 486.3346pt>
File: fig_reduced_dual_step4.png Graphic file (type png)
<use fig_reduced_dual_step4.png>
Package pdftex.def Info: fig_reduced_dual_step4.png used on input line 167.
Package pdftex.def Info: fig_reduced_dual_step4.png used on input line 168.
(pdftex.def) Requested size: 172.79846pt x 171.44409pt.
LaTeX Warning: `h' float specifier changed to `ht'.
@@ -226,17 +226,17 @@ uced_dual_step3.png> <./fig_reduced_dual_step4.png>]
<fig_chord_apex_step1.png, id=34, 505.03976pt x 502.06393pt>
File: fig_chord_apex_step1.png Graphic file (type png)
<use fig_chord_apex_step1.png>
Package pdftex.def Info: fig_chord_apex_step1.png used on input line 288.
Package pdftex.def Info: fig_chord_apex_step1.png used on input line 289.
(pdftex.def) Requested size: 251.9989pt x 250.5104pt.
<fig_chord_apex_step2.png, id=35, 490.16064pt x 499.51323pt>
File: fig_chord_apex_step2.png Graphic file (type png)
<use fig_chord_apex_step2.png>
Package pdftex.def Info: fig_chord_apex_step2.png used on input line 289.
Package pdftex.def Info: fig_chord_apex_step2.png used on input line 290.
(pdftex.def) Requested size: 172.79846pt x 176.08986pt.
<fig_chord_apex_step3.png, id=36, 490.16064pt x 499.51323pt>
File: fig_chord_apex_step3.png Graphic file (type png)
<use fig_chord_apex_step3.png>
Package pdftex.def Info: fig_chord_apex_step3.png used on input line 290.
Package pdftex.def Info: fig_chord_apex_step3.png used on input line 291.
(pdftex.def) Requested size: 172.79846pt x 176.08986pt.
@@ -244,7 +244,7 @@ LaTeX Warning: `h' float specifier changed to `ht'.
[4] [5 <./fig_chord_apex_step1.png> <./fig_chord_apex_step2.png> <./fig_chord_a
pex_step3.png>] [6]
Overfull \hbox (4.76643pt too wide) in paragraph at lines 432--439
Overfull \hbox (4.76643pt too wide) in paragraph at lines 433--440
\OT1/cmr/m/n/10 which $\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[
]\OT1/cmr/m/n/10 ) = \OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\
OT1/cmr/m/n/10 )$ and $\OML/cmm/m/it/10 '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[
@@ -253,31 +253,60 @@ T1/cmr/m/n/10 )\OML/cmm/m/it/10 ; '[]\OT1/cmr/m/n/10 (\OML/cmm/m/it/10 f[]\OT1/
cmr/m/n/10 )$
[]
<fig_alg_step0.png, id=49, 462.528pt x 440.42188pt>
<fig_alg_step0.png, id=49, 399.6106pt x 459.55217pt>
File: fig_alg_step0.png Graphic file (type png)
<use fig_alg_step0.png>
Package pdftex.def Info: fig_alg_step0.png used on input line 478.
(pdftex.def) Requested size: 115.20264pt x 109.69525pt.
<fig_alg_step1.png, id=50, 462.528pt x 440.42188pt>
Package pdftex.def Info: fig_alg_step0.png used on input line 479.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
<fig_alg_step1.png, id=50, 399.6106pt x 459.55217pt>
File: fig_alg_step1.png Graphic file (type png)
<use fig_alg_step1.png>
Package pdftex.def Info: fig_alg_step1.png used on input line 479.
(pdftex.def) Requested size: 115.20264pt x 109.69525pt.
<fig_alg_step2.png, id=51, 480.80806pt x 371.1277pt>
Package pdftex.def Info: fig_alg_step1.png used on input line 480.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
<fig_alg_step2.png, id=51, 399.6106pt x 459.55217pt>
File: fig_alg_step2.png Graphic file (type png)
<use fig_alg_step2.png>
Package pdftex.def Info: fig_alg_step2.png used on input line 480.
(pdftex.def) Requested size: 115.20264pt x 88.91956pt.
Package pdftex.def Info: fig_alg_step2.png used on input line 481.
(pdftex.def) Requested size: 115.20264pt x 132.48134pt.
Underfull \hbox (badness 4391) in paragraph at lines 498--498
\OT1/cmr/m/sc/10 Figure 3.\OT1/cmr/m/n/10 Algorithm 3.1[] on $\OML/cmm/m/it/10
G[] \OT1/cmr/m/n/10 = [](\OML/cmm/m/it/10 G\OT1/cmr/m/n/10 )$, where $\OML/cmm/
m/it/10 G$
[]
Underfull \hbox (badness 3623) in paragraph at lines 498--498
\OT1/cmr/m/n/10 is the first min-degree-$5$ plantri tri-an-gu-la-tion on $14$ v
er-
[]
Underfull \hbox (badness 3179) in paragraph at lines 498--498
\OT1/cmr/m/n/10 tices and $\OML/cmm/m/it/10 '[]$ \OT1/cmr/m/n/10 is a spe-cific
proper $3$-edge-colouring of $\OML/cmm/m/it/10 H[]$
[]
Underfull \hbox (badness 3209) in paragraph at lines 498--498
\OT1/cmr/m/n/10 that sat-is-fies both the chord-apex con-di-tion (Lemma 2.6[])
[]
Underfull \hbox (badness 6094) in paragraph at lines 498--498
\OT1/cmr/m/n/10 and the Kempe-cycle con-di-tion (Lemma 2.7[]), found by
[]
[7] [8 <./fig_alg_step0.png> <./fig_alg_step1.png> <./fig_alg_step2.png>]
(./paper.aux) )
Here is how much of TeX's memory you used:
3075 strings out of 478268
43781 string characters out of 5846347
343216 words of memory out of 5000000
21112 multiletter control sequences out of 15000+600000
3079 strings out of 478268
43848 string characters out of 5846347
344259 words of memory out of 5000000
21116 multiletter control sequences out of 15000+600000
476532 words of font info for 56 fonts, out of 8000000 for 9000
1302 hyphenation exceptions out of 8191
69i,8n,76p,918b,298s stack positions out of 10000i,1000n,20000p,200000b,200000s
69i,8n,76p,1306b,298s stack positions out of 10000i,1000n,20000p,200000b,200000s
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
m/cmcsc10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/c
@@ -289,15 +318,16 @@ mi7.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr1
fb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb>
</usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb></u
sr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></us
r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/
local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/l
ocal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/loc
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/loca
l/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (8 pages, 983037 bytes).
r/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy5.pfb></usr/
local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/lo
cal/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb></usr/loc
al/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti8.pfb></usr/local
/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/
texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb>
Output written on paper.pdf (8 pages, 969703 bytes).
PDF statistics:
132 PDF objects out of 1000 (max. 8388607)
69 compressed objects within 1 object stream
137 PDF objects out of 1000 (max. 8388607)
72 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000)
51 words of extra memory for PDF output out of 10000 (max. 10000000)