Add contraction-lift proof-strategy sketch toward Conjecture 5.7
Section 5.6 sketches an inductive route to the simple-resolution md4 surjectivity conjecture: - Lemma 5.8 (good contraction): every md4 triangulation on n >= 7 vertices has a degree-4 vertex with an md4-preserving diagonal contraction. Empirically true at n=7..11; proof obligation called out. - Lemma 5.9 (lift): given a labelled preimage of the contracted triangulation, reinserting the contracted vertex at the diagonal-bounded quadrilateral yields a preimage of the original triangulation. Proof obligation called out. - Inductive scheme paragraph chains the two lemmas with the octahedron at n=6 as the base case, citing the n=7 hand-verification (already scripted in experiments/inductive_lift_check.py). Lemmas are stated without proof; the three remaining proof obligations are explicit. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -54,18 +54,23 @@
|
||||
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Coverage test for Conjecture\nonbreakingspace \ref {conj:simple-md4}.}}{6}{section*.2}\protected@file@percent }
|
||||
\newlabel{obs:md4-simple-resolution}{{6.6}{7}{}{theorem.6.6}{}}
|
||||
\newlabel{conj:simple-md4}{{6.7}{7}{Simple-resolution $\mathrm {md}_4$ surjectivity}{theorem.6.7}{}}
|
||||
\newlabel{q:terminate-all-n}{{6.8}{7}{}{theorem.6.8}{}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Discussion and open questions}}{7}{section.7}\protected@file@percent }
|
||||
\newlabel{sec:contraction-lift}{{6.7}{7}{Towards a proof: a contraction--lift strategy}{subsection.6.7}{}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{6.7}{Towards a proof: a contraction--lift strategy}}{7}{subsection.6.7}\protected@file@percent }
|
||||
\newlabel{lem:good-contraction}{{6.8}{7}{Good contraction}{theorem.6.8}{}}
|
||||
\newlabel{lem:lift}{{6.9}{7}{Lift}{theorem.6.9}{}}
|
||||
\@writefile{toc}{\contentsline {paragraph}{\tocparagraph {}{}{Inductive scheme.}}{8}{section*.3}\protected@file@percent }
|
||||
\newlabel{q:terminate-all-n}{{6.10}{8}{}{theorem.6.10}{}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{Discussion and open questions}}{8}{section.7}\protected@file@percent }
|
||||
\bibcite{appelhaken}{1}
|
||||
\bibcite{rsst}{2}
|
||||
\bibcite{tutte}{3}
|
||||
\bibcite{chartrand}{4}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{12.7778pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent0}{13.28882pt}
|
||||
\newlabel{tocindent1}{18.3999pt}
|
||||
\newlabel{tocindent2}{29.38873pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Implementation}}{8}{section.8}\protected@file@percent }
|
||||
\newlabel{sec:impl}{{8}{8}{Implementation}{section.8}{}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{8}{section*.3}\protected@file@percent }
|
||||
\gdef \@abspage@last{8}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Implementation}}{9}{section.8}\protected@file@percent }
|
||||
\newlabel{sec:impl}{{8}{9}{Implementation}{section.8}{}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{section*.4}\protected@file@percent }
|
||||
\gdef \@abspage@last{9}
|
||||
|
||||
@@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 20 MAY 2026 13:40
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 20 MAY 2026 13:47
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
%&-line parsing enabled.
|
||||
@@ -363,27 +363,18 @@ Package hyperref Warning: Token not allowed in a PDF string (Unicode):
|
||||
Package hyperref Warning: Token not allowed in a PDF string (Unicode):
|
||||
(hyperref) removing `math shift' on input line 370.
|
||||
|
||||
[5] [6] [7] [8] (./paper.aux)
|
||||
|
||||
LaTeX Warning: Label(s) may have changed. Rerun to get cross-references right.
|
||||
|
||||
|
||||
Package rerunfilecheck Warning: File `paper.out' has changed.
|
||||
(rerunfilecheck) Rerun to get outlines right
|
||||
(rerunfilecheck) or use package `bookmark'.
|
||||
|
||||
Package rerunfilecheck Info: Checksums for `paper.out':
|
||||
(rerunfilecheck) Before: 00D0EBD21A3E3804EF6FF0D2A45466A8;3956
|
||||
(rerunfilecheck) After: C580028718693FDA81B49B22CE461AD9;3956.
|
||||
[5] [6] [7] [8] [9] (./paper.aux)
|
||||
Package rerunfilecheck Info: File `paper.out' has not changed.
|
||||
(rerunfilecheck) Checksum: B89F5E80B6733BD762B0E7C56A5250F0;4280.
|
||||
)
|
||||
Here is how much of TeX's memory you used:
|
||||
8945 strings out of 478268
|
||||
138142 string characters out of 5846347
|
||||
440795 words of memory out of 5000000
|
||||
26856 multiletter control sequences out of 15000+600000
|
||||
8956 strings out of 478268
|
||||
138278 string characters out of 5846347
|
||||
440946 words of memory out of 5000000
|
||||
26860 multiletter control sequences out of 15000+600000
|
||||
475834 words of font info for 54 fonts, out of 8000000 for 9000
|
||||
1302 hyphenation exceptions out of 8191
|
||||
69i,9n,76p,396b,466s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
69i,9n,76p,396b,533s 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/cm/cmcsc10.pfb
|
||||
></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb>
|
||||
@@ -402,10 +393,10 @@ ve/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy6.pfb></usr/local/texlive
|
||||
/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb></usr/local/texlive/2022/
|
||||
texmf-dist/fonts/type1/public/amsfonts/symbols/msam10.pfb></usr/local/texlive/2
|
||||
022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
|
||||
Output written on paper.pdf (8 pages, 258447 bytes).
|
||||
Output written on paper.pdf (9 pages, 270602 bytes).
|
||||
PDF statistics:
|
||||
281 PDF objects out of 1000 (max. 8388607)
|
||||
232 compressed objects within 3 object streams
|
||||
60 named destinations out of 1000 (max. 500000)
|
||||
145 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
310 PDF objects out of 1000 (max. 8388607)
|
||||
260 compressed objects within 3 object streams
|
||||
67 named destinations out of 1000 (max. 500000)
|
||||
153 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
@@ -13,6 +13,7 @@
|
||||
\BOOKMARK [2][-]{subsection.6.4}{\376\377\0006\000.\0004\000.\000\040\000P\000h\000a\000s\000e\000\040\0002\000:\000\040\000o\000u\000t\000e\000r\000-\000i\000n\000c\000i\000d\000e\000n\000t\000\040\000f\000a\000c\000e\000s}{section.6}% 13
|
||||
\BOOKMARK [2][-]{subsection.6.5}{\376\377\0006\000.\0005\000.\000\040\000S\000i\000m\000p\000l\000e\000\040\000l\000e\000v\000e\000l\000\040\000r\000e\000s\000o\000l\000u\000t\000i\000o\000n\000s}{section.6}% 14
|
||||
\BOOKMARK [2][-]{subsection.6.6}{\376\377\0006\000.\0006\000.\000\040\000E\000m\000p\000i\000r\000i\000c\000a\000l\000\040\000s\000t\000a\000t\000u\000s}{section.6}% 15
|
||||
\BOOKMARK [1][-]{section.7}{\376\377\0007\000.\000\040\000D\000i\000s\000c\000u\000s\000s\000i\000o\000n\000\040\000a\000n\000d\000\040\000o\000p\000e\000n\000\040\000q\000u\000e\000s\000t\000i\000o\000n\000s}{}% 16
|
||||
\BOOKMARK [1][-]{section.8}{\376\377\0008\000.\000\040\000I\000m\000p\000l\000e\000m\000e\000n\000t\000a\000t\000i\000o\000n}{}% 17
|
||||
\BOOKMARK [1][-]{section*.3}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 18
|
||||
\BOOKMARK [2][-]{subsection.6.7}{\376\377\0006\000.\0007\000.\000\040\000T\000o\000w\000a\000r\000d\000s\000\040\000a\000\040\000p\000r\000o\000o\000f\000:\000\040\000a\000\040\000c\000o\000n\000t\000r\000a\000c\000t\000i\000o\000n\040\023\000l\000i\000f\000t\000\040\000s\000t\000r\000a\000t\000e\000g\000y}{section.6}% 16
|
||||
\BOOKMARK [1][-]{section.7}{\376\377\0007\000.\000\040\000D\000i\000s\000c\000u\000s\000s\000i\000o\000n\000\040\000a\000n\000d\000\040\000o\000p\000e\000n\000\040\000q\000u\000e\000s\000t\000i\000o\000n\000s}{}% 17
|
||||
\BOOKMARK [1][-]{section.8}{\376\377\0008\000.\000\040\000I\000m\000p\000l\000e\000m\000e\000n\000t\000a\000t\000i\000o\000n}{}% 18
|
||||
\BOOKMARK [1][-]{section*.4}{\376\377\000R\000e\000f\000e\000r\000e\000n\000c\000e\000s}{}% 19
|
||||
|
||||
Binary file not shown.
@@ -539,6 +539,102 @@ vertices is not reachable by the algorithm; at $n = 10$, two further
|
||||
iso-classes with four degree-$3$ vertices and high-degree hubs fail to
|
||||
appear among algorithm outputs.
|
||||
|
||||
\subsection{Towards a proof: a contraction--lift strategy}
|
||||
\label{sec:contraction-lift}
|
||||
|
||||
We sketch an inductive strategy for Conjecture~\ref{conj:simple-md4}
|
||||
that we have verified empirically at small $n$ and offer here as a
|
||||
roadmap for further work.
|
||||
|
||||
Let $T$ be a plane triangulation with minimum degree at least $4$, and
|
||||
let $v \in V(T)$ be a degree-$4$ vertex with cyclic neighbors
|
||||
$a, b, c, d$ (in the cyclic order inherited from $T$'s planar
|
||||
embedding). Removing $v$ from $T$ exposes the $4$-cycle $abcd$, which we
|
||||
retriangulate by adding one of the two diagonals $(a, c)$ or $(b, d)$.
|
||||
We call this operation \emph{contraction at $v$ along diagonal
|
||||
$(a, c)$}, denoted $T_{v, (a, c)}$. The contraction is \emph{valid} when
|
||||
the chosen diagonal is not already an edge of $T$.
|
||||
|
||||
\begin{lemma}[Good contraction]
|
||||
\label{lem:good-contraction}
|
||||
Let $T$ be a plane triangulation on $n \geq 7$ vertices with minimum
|
||||
degree at least $4$. Then there exist a degree-$4$ vertex $v \in V(T)$,
|
||||
with cyclic neighbors $a, b, c, d$, and an unordered pair
|
||||
$\{a, c\}$ such that:
|
||||
\begin{enumerate}
|
||||
\item $(a, c) \not\in E(T)$;
|
||||
\item $\deg_T(b) \geq 5$ and $\deg_T(d) \geq 5$.
|
||||
\end{enumerate}
|
||||
Under these conditions $T_{v, (a, c)}$ is a plane triangulation on
|
||||
$n - 1$ vertices with minimum degree at least $4$.
|
||||
\end{lemma}
|
||||
|
||||
The conditions of Lemma~\ref{lem:good-contraction} ensure that the
|
||||
contraction is valid (1) and md$_4$-preserving (2): the only vertices
|
||||
whose degree changes under $T \to T_{v, (a, c)}$ are $a, b, c, d$, with
|
||||
$\deg(a)$ and $\deg(c)$ unchanged (each loses the edge to $v$ but gains
|
||||
the edge from the diagonal), while $\deg(b)$ and $\deg(d)$ each decrease
|
||||
by $1$.
|
||||
|
||||
The lemma is empirically true at $n = 7, \ldots, 11$ for every md$_4$
|
||||
iso-class; we conjecture it holds for all $n \geq 7$. The $n = 6$ case
|
||||
is excluded: the unique md$_4$ iso-class is the octahedron, in which
|
||||
every vertex has all four cyclic neighbors at degree $4$ and so no
|
||||
contraction preserves md$_4$. The octahedron is therefore the base case
|
||||
of the proposed induction.
|
||||
|
||||
\begin{lemma}[Lift]
|
||||
\label{lem:lift}
|
||||
Let $T$ be a plane triangulation with minimum degree at least $4$, and
|
||||
suppose Lemma~\ref{lem:good-contraction} applies via vertex $v$ and
|
||||
diagonal $(a, c)$ with $T_{v, (a, c)}$ the resulting contraction. Let
|
||||
$H$ be a plane triangulation on $V(T_{v, (a, c)}) = V(T) \setminus \{v\}$
|
||||
and $S$ a level source of $H$ such that the algorithm of
|
||||
Section~\ref{sec:flip-algorithm} applied to $(H, S)$ produces
|
||||
$T_{v, (a, c)}$ as a labelled simple graph. Define the \emph{lift}
|
||||
$G \;:=\; H[a, b, c, d, v]$ by:
|
||||
\begin{itemize}
|
||||
\item adding vertex $v$ to $V(H)$;
|
||||
\item removing the edge $(a, c)$ from $E(H)$;
|
||||
\item adding the four edges $(v, a), (v, b), (v, c), (v, d)$.
|
||||
\end{itemize}
|
||||
Then $G$ is a plane triangulation on $|V(T)|$ vertices, and the
|
||||
algorithm of Section~\ref{sec:flip-algorithm} applied to $(G, S)$
|
||||
produces $T$.
|
||||
\end{lemma}
|
||||
|
||||
Lemma~\ref{lem:lift} requires that $(a, c) \in E(H)$ and that the two
|
||||
triangles of $H$ bordering $(a, c)$ have boundary
|
||||
$\{a, b, c, d\}$. When these conditions hold, the lift restores a
|
||||
degree-$4$ vertex $v$ inserted into the quadrilateral $abcd$; when they
|
||||
fail, the lift is undefined and a different labelled preimage $H$ must
|
||||
be chosen.
|
||||
|
||||
\paragraph{Inductive scheme.}
|
||||
Conjecture~\ref{conj:simple-md4} would follow from
|
||||
Lemmas~\ref{lem:good-contraction} and~\ref{lem:lift} together with the
|
||||
existence at each step of a labelled preimage $H$ satisfying the lift's
|
||||
side conditions. The base case is the octahedron at $n = 6$, which is
|
||||
empirically a simple level resolution
|
||||
(Observation~\ref{obs:md4-simple-resolution}). The inductive step takes
|
||||
an md$_4$ target $T$ on $n$ vertices, applies
|
||||
Lemma~\ref{lem:good-contraction} to obtain an md$_4$ contraction
|
||||
$T_{v, (a, c)}$ on $n - 1$ vertices, invokes the inductive hypothesis to
|
||||
produce a labelled preimage $H$, and applies Lemma~\ref{lem:lift} to
|
||||
lift $H$ to $G$ with $\mathrm{alg}(G, S) = T$.
|
||||
|
||||
We have verified the entire scheme by hand for the unique md$_4$
|
||||
iso-class at $n = 7$: contraction at $v = 2$ along diagonal $(4, 3)$
|
||||
yields the octahedron on six vertices labelled $\{0, 1, 3, 4, 5, 6\}$;
|
||||
a labelled preimage $H$ exists with source $S = \{0, 1, 6\}$; lifting
|
||||
along $(4, 3, v = 2)$ produces a triangulation $G$ on seven vertices on
|
||||
which the algorithm with source $S$ recovers $T$ exactly. The principal
|
||||
remaining work is a proof of Lemma~\ref{lem:good-contraction} for all
|
||||
$n \geq 7$, a proof of Lemma~\ref{lem:lift} (which involves analysing
|
||||
how the algorithm's depth-guided flips interact with the added vertex
|
||||
$v$), and a guarantee that a label-faithful preimage $H$ always
|
||||
exists.
|
||||
|
||||
\begin{question}
|
||||
\label{q:terminate-all-n}
|
||||
Does Phase~1 terminate for all $(G, S)$? Equivalently, is there an
|
||||
|
||||
Reference in New Issue
Block a user