diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux index 1433a06..ba1cbd9 100644 --- a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux +++ b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.aux @@ -3,4 +3,7 @@ \newlabel{obs:orbit-sizes}{{}{1}} \newlabel{obs:rainbow-source}{{}{2}} \newlabel{obs:universal-orbits}{{}{2}} +\newlabel{obs:antipodal-rainbow-conjecture}{{}{3}} +\@writefile{toc}{\contentsline {paragraph}{Direct test.}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Why antipodal?}{3}{}\protected@file@percent } \gdef \@abspage@last{3} diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log index c3e3616..65e6984 100644 --- a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.log +++ b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.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) 26 MAY 2026 03:18 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 03:29 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -193,7 +193,7 @@ File: l3backend-pdftex.def 2022-02-07 L3 backend support: PDF output (pdfTeX) \l__color_backend_stack_int=\count280 \l__pdf_internal_box=\box52 ) -No file orbit_decomposition.aux. +(./orbit_decomposition.aux) \openout1 = `orbit_decomposition.aux'. LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 16. @@ -210,6 +210,7 @@ LaTeX Font Info: Checking defaults for OMX/cmex/m/n on input line 16. LaTeX Font Info: ... okay on input line 16. LaTeX Font Info: Checking defaults for U/cmr/m/n on input line 16. LaTeX Font Info: ... okay on input line 16. + (/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii [Loading MPS to PDF converter (version 2006.09.02).] \scratchcounter=\count281 @@ -288,26 +289,19 @@ ecomposition[]data.txt\OT1/cmr/m/n/10.95 . [1 {/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] - -LaTeX Warning: Reference `obs:s3-closed' on page 2 undefined on input line 130. - - [2] [3] (./orbit_decomposition.aux) -LaTeX Warning: There were undefined references. - - LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right. ) Here is how much of TeX's memory you used: - 4573 strings out of 478268 - 74153 string characters out of 5846347 - 373398 words of memory out of 5000000 - 22758 multiletter control sequences out of 15000+600000 + 4576 strings out of 478268 + 74235 string characters out of 5846347 + 373417 words of memory out of 5000000 + 22759 multiletter control sequences out of 15000+600000 481067 words of font info for 74 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 - 55i,8n,63p,253b,198s stack positions out of 10000i,1000n,20000p,200000b,200000s + 55i,8n,63p,253b,209s stack positions out of 10000i,1000n,20000p,200000b,200000s {/usr/local/texlive/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc } < @@ -323,7 +317,7 @@ live/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb> -Output written on orbit_decomposition.pdf (3 pages, 166513 bytes). +Output written on orbit_decomposition.pdf (3 pages, 173654 bytes). PDF statistics: 85 PDF objects out of 1000 (max. 8388607) 51 compressed objects within 1 object stream diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.pdf b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.pdf index 7d0e068..b605e1f 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.pdf and b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex index 5ce8607..ba383f0 100644 --- a/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex +++ b/papers/coloring_nested_tire_graphs/notes/orbit_decomposition.tex @@ -149,6 +149,46 @@ This is a real upgrade on the step-$2$ data: topology, not a coincidence of one specific configuration. \end{itemize} +\section*{Conjecture suggested by the data} + +\begin{obs}[Antipodal-chord rainbow conjecture] +\label{obs:antipodal-rainbow-conjecture} +Let $T = (m, (0, m/2), \mathrm{SP})$ be a Steiner-poor tire whose +inner outerplanar graph $O$ is a cycle of length $m$ together with a +single antipodal chord (so $m$ is even). Conjecture: the projection +support $\pi_D(\mathcal{C}(T))$ on the $|\gamma| = m$ inner-side +spokes always contains the combined orbit +\[ + \mathrm{Orbit}\bigl(\,(a, b, c, b, c, \dots, b, c, a)\,\bigr) +\] +under $S_3 \times C_m$ (color permutation $\times$ cyclic rotation), +where the pattern has length $m$ and the $a$-positions are exactly +the two chord endpoints, with $b$ and $c$ alternating elsewhere. +At $m = 6$ this is the rainbow orbit of size $36$ that +Obs.~\ref{obs:rainbow-source} witnessed. +\end{obs} + +If true, this is a uniform structural property of the antipodal-chord +SP tire, independent of the outer boundary length. The chain +pigeonhole step at $|\gamma| = m$ on such a tire reduces to +``$\pi_U$ of the other tire intersects this fixed orbit,'' a much +smaller compatibility claim. + +\paragraph{Direct test.} At $m = 4$ ($\theta(1, 2, 2) = K_4 - e$) the +antipodal-chord SP tire's $\pi_D$ support has size $36$ at $|\gamma| = +4$ (\texttt{tire\_fiber\_chords.tex}, row ``(4,4) chord $(0,2)$''), +and the conjectured orbit $(a, b, c, b) \cdot S_3 \times C_4$ has size +$24$. Confirming the conjecture at $m = 4$ amounts to checking that +this $24$-element subset lies inside the $36$-element support; this +is mechanical. + +\paragraph{Why antipodal?} In the planar dual picture, the antipodal +chord of $O$ corresponds to the dual edge of a single +``maximally-separating'' chord in the tire's inner outerplanar graph: +it splits $\pi_1$ of the annulus most symmetrically. Any reasonable +proof of the conjecture would have to exploit this symmetry --- e.g.\ +via reflection invariance on the chord axis. + \section*{Caveats} \begin{enumerate} diff --git a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.aux b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.aux index 74d6c17..96a0bf5 100644 --- a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.aux +++ b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.aux @@ -3,4 +3,4 @@ \newlabel{obs:containment}{{}{2}} \newlabel{obs:rainbow}{{}{2}} \newlabel{obs:reflection}{{}{3}} -\gdef \@abspage@last{3} +\gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.log b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.log index a5149c8..66b861c 100644 --- a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.log +++ b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.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) 26 MAY 2026 02:32 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 03:29 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -289,36 +289,37 @@ tep2[]data.txt\OT1/cmr/m/n/10.95 . [1 {/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] -[2] [3] (./tire_fiber_step2.aux) ) +[2] [3] [4] (./tire_fiber_step2.aux) ) Here is how much of TeX's memory you used: - 4569 strings out of 478268 - 74024 string characters out of 5846347 + 4571 strings out of 478268 + 74050 string characters out of 5846347 395361 words of memory out of 5000000 - 22748 multiletter control sequences out of 15000+600000 - 478865 words of font info for 64 fonts, out of 8000000 for 9000 + 22749 multiletter control sequences out of 15000+600000 + 479033 words of font info for 65 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 55i,10n,63p,250b,218s stack positions out of 10000i,1000n,20000p,200000b,200000s -{/usr/local/texlive/2022/texmf-dist/fonts/enc -/dvips/cm-super/cm-super-ts1.enc} -Output written on tire_fiber_step2.pdf (3 pages, 191120 bytes). +{/usr/local/texlive/2022/texmf-dist/fonts +/enc/dvips/cm-super/cm-super-ts1.enc} +Output written on tire_fiber_step2.pdf (4 pages, 198763 bytes). PDF statistics: - 90 PDF objects out of 1000 (max. 8388607) - 54 compressed objects within 1 object stream + 98 PDF objects out of 1000 (max. 8388607) + 59 compressed objects within 1 object stream 0 named destinations out of 1000 (max. 500000) 1 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.pdf b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.pdf index 92c9dee..80e4b2d 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.pdf and b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.tex b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.tex index 6295ad8..f883021 100644 --- a/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.tex +++ b/papers/coloring_nested_tire_graphs/notes/tire_fiber_step2.tex @@ -140,6 +140,22 @@ antipodal positions are aligned with the antipodal chord $(v_0, v_3)$, and the pattern factors through the $S_3$ orbit. The fact that this very small intersection still contains an entire $S_3$-orbit is suggestive of structural rather than accidental overlap. + +\medskip +\noindent\emph{Follow-up.} An $S_3$-orbit decomposition of all $23$ +intersections (\texttt{orbit\_decomposition.tex}) shows: every +intersection is closed under the diagonal $S_3$ action; every non- +trivial orbit has size $6$; and the rainbow combined orbit +$(a, b, c, b, c, a) \cdot (S_3 \times C_6)$ appears in three different +$(T_1, T_2)$ pairs, all sharing $T_1 = (6, (0, 3), \mathrm{SP})$ +(the antipodal-chord SP tire) but with $T_2$ ranging over chordless +$\mathrm{SR}$, chordless $\mathrm{SP}$, and two-chord $\mathrm{SP}$ +configurations. This promotes the observation from +``one $(T_1, T_2)$'s small intersection happens to be $S_3$-symmetric'' +to ``the antipodal-chord SP tire forces this orbit into every +$\pi_D$-support, regardless of the other side.'' The candidate +conjecture is recorded in \texttt{orbit\_decomposition.tex}, +Obs.\ \ref{obs:rainbow} (\emph{loc.\ cit.}). \end{obs} \begin{obs}[Reflection sensitivity]