diff --git a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux index 606a1d8..638b45d 100644 --- a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux +++ b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.aux @@ -5,10 +5,13 @@ \@writefile{toc}{\contentsline {paragraph}{Translating to primal cycles.}{1}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Is this realisable?}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{However:}{2}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{Next layer.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Forced $12$ degree-$5$ vertices.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Second-link length.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Second-link length does \emph {not} pin down min cyclic cut.}{2}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{Conclusion of this section.}{3}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Relevance to the cut-tire framework.}{3}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Yes, in principle.}{3}{}\protected@file@percent } -\@writefile{toc}{\contentsline {paragraph}{What would force this?}{3}{}\protected@file@percent } +\@writefile{toc}{\contentsline {paragraph}{What would force this?}{4}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{The framework adapts.}{4}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Bottom line.}{4}{}\protected@file@percent } \gdef \@abspage@last{4} diff --git a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.log b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.log index d1a8fea..0b3b581 100644 --- a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.log +++ b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.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) 27 MAY 2026 00:22 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 27 MAY 2026 00:25 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -265,33 +265,34 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B [4] (./even_separating_cycle.aux) ) Here is how much of TeX's memory you used: - 3257 strings out of 478268 - 48510 string characters out of 5846347 - 348660 words of memory out of 5000000 - 21447 multiletter control sequences out of 15000+600000 - 480899 words of font info for 73 fonts, out of 8000000 for 9000 + 3259 strings out of 478268 + 48537 string characters out of 5846347 + 347660 words of memory out of 5000000 + 21448 multiletter control sequences out of 15000+600000 + 481258 words of font info for 74 fonts, out of 8000000 for 9000 1141 hyphenation exceptions out of 8191 55i,5n,62p,238b,198s stack positions out of 10000i,1000n,20000p,200000b,200000s {/usr/local/texlive/2022/texmf-dist/fonts/enc/dv ips/cm-super/cm-super-ts1.enc} -Output written on even_separating_cycle.pdf (4 pages, 174458 bytes). +lic/amsfonts/cm/cmbxti10.pfb> +Output written on even_separating_cycle.pdf (4 pages, 185871 bytes). PDF statistics: - 88 PDF objects out of 1000 (max. 8388607) - 53 compressed objects within 1 object stream + 93 PDF objects out of 1000 (max. 8388607) + 56 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/even_separating_cycle.pdf b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.pdf index a1f5150..c12b252 100644 Binary files a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.pdf and b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.pdf differ diff --git a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.tex b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.tex index 98838c1..f606f45 100644 --- a/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.tex +++ b/papers/coloring_nested_tire_graphs/notes/even_separating_cycle.tex @@ -106,41 +106,51 @@ possible in principle. examples (icosahedron's dual = dodecahedron, etc.) all have many even cuts of size $6$ as well as odd cuts. -\section*{A heuristic suggesting ``yes'' --- vertex links} +\section*{What the maximal-planar constraint forces} -Every vertex $v$ in a planar triangulation $G$ has a \emph{link}: -the cycle formed by its neighbours. By Birkhoff, this link is a -$5$-cycle isolating $v$. In $G^*$, this corresponds to a -$5$-edge cut isolating a single triangle face of $G^*$ (= one -vertex of $G$). +Maximal planar (= triangulation) gives strong constraints: +$E = 3V - 6$, $F = 2V - 4$, $\sum \deg = 6V - 12$. Average degree +approaches $6$ from below. Internally $6$-connected forces min +degree $\ge 5$. -\paragraph{Next layer.} Consider the ``second link'' of $v$: the -set of vertices at $G$-distance exactly $2$ from $v$. This forms -a cycle around the link, of length depending on the degrees of -link vertices. +\paragraph{Forced $12$ degree-$5$ vertices.} If all degrees are in +$\{5, 6\}$: +\[ +5 n_5 + 6 n_6 = 6V - 12, \quad n_5 + n_6 = V \implies n_5 = 12. +\] +So exactly $12$ degree-$5$ vertices, with the remaining $V - 12$ +of degree $6$. For $V > 12$, at least some degree-$5$ vertex has a +degree-$6$ link neighbour. -If all link vertices have degree $5$ (= minimum for internally -$6$-connected): the link's $5$ vertices contribute $5 \cdot 5 = 25$ -incidences, of which $5$ are to $v$ (the centre) and $2 \cdot 5 -= 10$ are between link vertices (the $5$-cycle). The remaining -$25 - 5 - 10 = 10$ incidences go to second-link vertices. But the -$5$ link vertices share their second-link vertices in pairs (each -triangle face containing $v$, a link vertex, and a second-link -vertex), so the second link has $\le 5$ distinct vertices. +\paragraph{Second-link length.} For vertex $v$ of degree $d$ in a +triangulation, the second link (cycle of vertices at distance +exactly $2$) has length +\[ +L_2(v) = d + \sum_{u \in \mathrm{link}(v)} (\deg u - 5). +\] +(Each link vertex contributes one ``shared'' boundary vertex with +each link-cycle neighbour, plus $\deg u - 5$ private boundary +vertices inside $u$'s fan.) For the icosahedron ($d = 5$, all +link degrees $5$): $L_2 = 5$. -Working through Euler more carefully: if all degrees are $\ge 5$ -and the link of $v$ has length $5$, the second link has length -exactly $5 \cdot (5 - 4) = 5$ (in the icosahedron, second link = -link of antipodal vertex). But in larger triangulations -(degrees of link vertices $\ge 5$, some higher), the second link -is generically a cycle of length $\sum_{u \in \text{link}}(\deg(u) -- 4) = 5 + \sum_u(\deg u - 5) \ge 5$ with equality only when all -link degrees are exactly $5$ (icosahedron case). +\paragraph{Second-link length does \emph{not} pin down min cyclic +cut.} The pentakis dodecahedron ($V = 32$): every degree-$5$ +vertex has all $5$ link vertices of degree $6$, so $L_2 = 5 + 5 = +10$ around degree-$5$ vertices. Around degree-$6$ vertices, +$L_2 = 6 + 0 \cdot 2 + 1 \cdot 4 = 10$. So no second-link +separator has length $6$. -So in larger internally $6$-connected triangulations, second-link -length $\ge 6$, often equal to $6$, often even (especially in -``vertex-transitive enough'' graphs). This is a heuristic for why -$6$-cycle separators are abundant, but it's not a proof. +\emph{But} the pentakis dodecahedron's dual (Buckminsterfullerene +graph) \emph{does} have $6$-edge cyclic cuts --- they arise as +separations \emph{not} surrounding a single vertex. So +second-link length is just one source of cyclic cuts; longer +constructions can yield smaller cuts. + +\paragraph{Conclusion of this section.} The maximal-planar +constraint forces some structural relations but does not pin down +the minimum non-facial cyclic cut to any specific value. A +minimum $4$CT counterexample could in principle have any min cut +size $\ge 6$. \section*{What I can conclude}