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dual_decomposition: pentagonal externals lemma + edge naming

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- Name the edges of the reduced-dual construction (merged, spike, sides)
  via a new definition; use these names in lem:chord-apex.
- Add lem:pentagonal-externals with full exhaustive proof: any proper
  3-edge-colouring near a pentagonal face of a cubic plane graph has its
  five external edges forming, up to cyclic rotation, the pattern
  (a, b, c, c, c) with {a, b, c} = {1, 2, 3} (iff).
- Cite the new lemma in the chord-apex proof scaffold as the lifting step.
- Remove the icosahedron experimental remark.

Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-22 20:22:00 -04:00
parent 20f19f0869
commit 0303225f39
6 changed files with 165 additions and 34 deletions
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papers/dual_decomposition_minimal_counterexamples/paper.aux
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\newlabel{def:minimal}{{1.2}{1}}
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\label{fig:reduced-dual-steps}
\end{figure}
\begin{definition}[Edges of the reduced dual]
\label{def:edge-names}
The four edges added in steps (3) and (4) of Definition~\ref{def:reduced-dual}
are named as follows. The chord $A_{i+3}A_{i+4}$ is the \emph{merged edge}; the
edge $A_{i+1}v_n$ is the \emph{spike edge}; and the edges $A_iv_n$ and
$A_{i+2}v_n$ are the \emph{side edges}. In the $i = 0$ case of
Figure~\ref{fig:reduced-dual-steps} these are $\{A_3, A_4\}$, $\{A_1, v_n\}$,
and $\{A_0, v_n\}, \{A_2, v_n\}$ respectively.
\end{definition}
We will use the following structural fact about proper $3$-edge-colourings near
a pentagonal face of a cubic plane graph; it is stated for a generic such graph
$H$, not specifically for the reduced dual.
\begin{lemma}[Pentagonal externals]
\label{lem:pentagonal-externals}
Let $H$ be a cubic plane graph and $F$ a pentagonal face of $H$, with
$\partial F$ traversed clockwise as $u_0, u_1, u_2, u_3, u_4$. For each $i$
let $f_i$ be the unique edge of $H$ incident to $u_i$ that does not lie on
$\partial F$. An assignment $\varphi$ of colours from $\{1, 2, 3\}$ to the ten
edges incident to $\{u_0, \dots, u_4\}$ is proper at every $u_i$ if and only if
there is some index $j$ such that
\[
\varphi(f_j) = \varphi(f_{j+1}) = \varphi(f_{j+2})
\quad\text{and}\quad
\{\varphi(f_{j+3}), \varphi(f_{j+4})\}
= \{1, 2, 3\} \setminus \{\varphi(f_j)\},
\]
indices mod $5$.
\end{lemma}
\begin{proof}
Write $e_i = u_i u_{i+1}$ for the boundary edges of $\partial F$ (indices mod
$5$). A colouring $\varphi$ is proper at every $u_i$ if and only if at each
$u_i$ the three incident edges $e_{i-1}, e_i, f_i$ receive three distinct
colours; whenever this holds, $\varphi(f_i)$ is forced to be the unique colour
in $\{1, 2, 3\} \setminus \{\varphi(e_{i-1}), \varphi(e_i)\}$, and $\varphi$
restricts to a proper $3$-edge-colouring of the cycle $\partial F$.
\textbf{($\Rightarrow$)} The line graph of $\partial F$ is $C_5$, whose
maximum independent set has size $2$, so no colour appears more than twice on
$\partial F$; and since $\partial F$ is an odd cycle, all three colours appear.
The colour multiset on $(\varphi(e_0), \dots, \varphi(e_4))$ is therefore
$(2, 2, 1)$, with the singleton at a unique position. Cyclically shifting
indices we may place this position at $0$; let $c$ be the singleton colour.
The remaining four edges form the path $e_1 e_2 e_3 e_4$, which by propriety
alternates between the other two colours, so for some labelling
$\{a, b, c\} = \{1, 2, 3\}$,
\[
(\varphi(e_0), \varphi(e_1), \varphi(e_2), \varphi(e_3), \varphi(e_4))
= (c, a, b, a, b).
\]
Reading off the forced values of $\varphi(f_i)$,
\[
\varphi(f_0) = a, \quad
\varphi(f_1) = b, \quad
\varphi(f_2) = \varphi(f_3) = \varphi(f_4) = c,
\]
which is the lemma's pattern at $j = 2$ (the cyclic shift maps this back to
the corresponding $j$ in the original indexing). This case is the unique
proper $3$-edge-colouring of $\partial F$ up to cyclic shift and permutation of
$\{1, 2, 3\}$ (since $5 \cdot 3! = 30 = P(C_5, 3)$, the chromatic polynomial of
$C_5$ at $3$), so it exhausts every proper $\varphi$.
\textbf{($\Leftarrow$)} The lemma's hypothesis is invariant under cyclic
shifts of indices and under permutations of $\{1, 2, 3\}$, so we may assume
$j = 2$, $\varphi(f_2) = \varphi(f_3) = \varphi(f_4) = c$, $\varphi(f_0) = a$,
and $\varphi(f_1) = b$, with $\{a, b, c\} = \{1, 2, 3\}$. Propriety at $u_i$
and $u_{i+1}$ requires $\varphi(e_i) \notin \{\varphi(f_i), \varphi(f_{i+1})\}$,
which gives
\[
\varphi(e_0) = c, \quad
\varphi(e_1) = a, \quad
\varphi(e_2) \in \{a, b\}, \quad
\varphi(e_3) \in \{a, b\}, \quad
\varphi(e_4) = b.
\]
The remaining propriety condition $\varphi(e_{i-1}) \neq \varphi(e_i)$ holds
automatically at $u_0, u_1, u_4$, forces $\varphi(e_2) = b$ at $u_2$, and then
forces $\varphi(e_3) = a$ at $u_3$. The resulting triples
$(\varphi(e_{i-1}), \varphi(e_i), \varphi(f_i))$ at $u_0, u_1, u_2, u_3, u_4$
are
\[
(b, c, a), \quad (c, a, b), \quad (a, b, c), \quad (b, a, c), \quad (a, b, c),
\]
each a permutation of $\{1, 2, 3\}$, so $\varphi$ is proper at every $u_i$.
\end{proof}
\begin{remark}
The two-element condition $\{\varphi(f_{j+3}), \varphi(f_{j+4})\}
= \{1,2,3\}\setminus\{\varphi(f_j)\}$ cannot be dropped: a 3-colouring
satisfying $\varphi(f_j) = \varphi(f_{j+1}) = \varphi(f_{j+2})$ alone need not
extend, e.g.\ $(1,1,1,1,2)$.
\end{remark}
Since $\widehat{G}'_{v,i}$ is the dual of a triangulation on fewer vertices than
$G$, it is $3$-edge-colourable by the minimality of $G$. The following lemma
constrains every such colouring.
\begin{lemma}
\label{lem:chord-apex}
Let $G$ be a minimal counterexample, and let $\widehat{G}'_{v,i}$ be a reduced
dual of its dual $G'$. Then in every proper $3$-edge-colouring of
$\widehat{G}'_{v,i}$, the merged edge and the spike edge receive the same
colour.
\end{lemma}
\begin{proof}
% TODO. Intended argument: given a proper 3-edge-colouring of
% \widehat{G}'_{v,i}, the colour at each A_k of the new edge there (one of the
% three v_n-edges for k in {i, i+1, i+2}, or the chord for k in {i+3, i+4})
% determines the colour of f_k in any lift to G' --- it is the unique colour
% not used by the two original edges at A_k. Because the chord has a single
% colour, f_{i+3} and f_{i+4} agree. If the merged edge and the spike edge
% receive distinct colours, the three v_n-edges contribute three distinct
% colours to f_i, f_{i+1}, f_{i+2}, and the resulting f-vector has shape
% (X, Y, Z, W, W) with X, Y, Z distinct and W \neq Y. By
% Lemma~\ref{lem:pentagonal-externals} applied to G' at the face F_v, this
% extends to a proper 3-edge-colouring of G' --- contradicting that G' (the
% dual of the minimal counterexample G) is not 3-edge-colourable. Hence the
% merged edge and the spike edge must share a colour.
\end{proof}
\end{document}