didericis
03dcd7c2fa
- 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>
29 lines
3.4 KiB
TeX
29 lines
3.4 KiB
TeX
\relax
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\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{The minimal counterexample}}{1}{}\protected@file@percent }
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\newlabel{lem:triangulate}{{1.1}{1}}
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\newlabel{def:minimal}{{1.2}{1}}
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\newlabel{lem:mindeg}{{1.4}{1}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The reduced dual}}{2}{}\protected@file@percent }
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\newlabel{def:reduced-dual}{{2.1}{2}}
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\newlabel{def:edge-names}{{2.3}{2}}
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\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The four steps of Definition\nonbreakingspace 2.1\hbox {}, illustrated on $G' = $ the dodecahedron (dual of the icosahedron) with $F_v$ the inner pentagon and $i = 0$. Top left: delete the five boundary vertices of $F_v$, leaving five degree-$2$ vertices on a new face $F$. Top right: order them clockwise as $A_0,\dots ,A_4$. Bottom left: add $v_n$ joined to $A_0, A_1, A_2$. Bottom right: add the chord $A_3 A_4$, giving the cubic plane graph $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,0}$.}}{3}{}\protected@file@percent }
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\newlabel{fig:reduced-dual-steps}{{1}{3}}
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\newlabel{lem:pentagonal-externals}{{2.4}{3}}
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\newlabel{lem:chord-apex}{{2.6}{4}}
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\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The proof of Lemma\nonbreakingspace 2.6\hbox {}, illustrated for $i = 0$ on $G' = $ the dodecahedron. Top: under the assumption $W \neq Y$, propriety at $v_n$ forces $W \in \{X, Z\}$. Bottom: in either case the lift to $G'$ has externals satisfying the hypothesis of Lemma\nonbreakingspace 2.4\hbox {}, which colours $\partial F_v$ to extend $\psi $ to a proper $3$-edge-colouring of $G'$.}}{5}{}\protected@file@percent }
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\newlabel{fig:chord-apex-proof}{{2}{5}}
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\newlabel{lem:kempe-spike}{{2.7}{6}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{An iterated reduction}}{7}{}\protected@file@percent }
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\newlabel{alg:iterated-reduction}{{3.1}{7}}
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\newlabel{rem:alg-invariants}{{3.2}{7}}
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\newlabel{rem:alg-chord-apex}{{3.3}{7}}
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\newlabel{tocindent-1}{0pt}
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent3}{0pt}
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\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces Algorithm\nonbreakingspace 3.1\hbox {} on $G'=\mathrm {dual}(G)$, where $G$ is the first min-degree-$5$ plantri triangulation on $14$ vertices and $\varphi _1$ is a specific proper $3$-edge-colouring of $H_1$ that satisfies both the chord-apex condition (Lemma\nonbreakingspace 2.6\hbox {}) and the Kempe-cycle condition (Lemma\nonbreakingspace 2.7\hbox {}), found by \texttt {experiments/search\_kempe\_property.py}. \emph {Left:} $G'$ ($24$ vertices, $36$ edges) with the chosen pentagonal face shaded. \emph {Centre:} $H_1$ ($20$ vertices, $30$ edges) after step\nonbreakingspace (1) with $i_1 = 1$, $3$-edge-coloured by $\varphi _1$; the four edges around $v_n^{(1)}$ in $E$ are drawn thicker, and the spike and merged edges share the colour green. \emph {Right:} $H_2$ ($16$ vertices, $24$ edges) after step\nonbreakingspace (3) with $i_t = 3$; eight edges are protected, and the algorithm terminates one step later (no remaining safe pentagonal face in $H_2$). The generating script is \texttt {experiments/draw\_iterated\_reduction\_n14.py}; layouts are Tutte barycentric embeddings with the outer face picked to keep $v_n^{(1)}, v_n^{(2)}$ in the interior.}}{8}{}\protected@file@percent }
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\newlabel{fig:iterated-reduction-trace}{{3}{8}}
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\newlabel{conj:no-all-distinct-coloring}{{3.4}{8}}
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\gdef \@abspage@last{8}
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