coloring_nested_tire_graphs: conjecture sketch on universal nesting
NEW Conjecture 1.19 (universal nesting of tire-tread trees,
sketch):
For any two rooted trees of tire treads T_1 = T(G_1, S_1) and
T_2 = T(G_2, S_2), T_1 NESTS into T_2:
Choose any tire T in T_2 and any non-trivial bounded face f of
its inner outerplanar graph O^(T). Then there exists a maximal
planar graph G̃ with level source S̃ such that:
(N1) T(G̃, S̃) contains T_2 as a sub-tree.
(N2) The sub-tree rooted at the new child of T at face f is
isomorphic to T_1.
Informally: any tree of tire treads can be inserted into any
non-trivial face slot of any other tree of tire treads. The
class of trees of tire treads is closed under composition by
face-slot insertion.
Followed by Remark 1.20 motivating the conjecture:
- Compositional colourability: if 4-colourability of G̃ follows
from 4-colourability of G_1, G_2 via parent-child consistency
(Remark 1.18 / former tree-coloring-factorisation), then 4CT
propagates through nesting. A min 4CT counterexample would have
to be irreducible under such nesting.
- Universality: trees of tire treads become a "term algebra" for
decomposing plane triangulations; coloring arguments can be
inductive on this algebra.
Open subquestions in remark:
- Precise notion of "isomorphic as rooted trees of tire treads"
(combinatorial vs geometric vs up to embedding).
- Constructive description of G̃ from G_1, G_2, f.
- Compatibility with Birkhoff's internally 6-connected condition.
Page count: 12 → ~13.
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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@@ -950,6 +950,75 @@ This is the structural setup underlying the chain-pigeonhole
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program for tire treads.
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program for tire treads.
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\end{remark}
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\end{remark}
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\begin{conjecture}[Universal nesting of tire-tread trees, sketch]
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\label{conj:universal-nesting}
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For any two rooted trees of tire treads
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$\mathcal{T}_1 = \mathcal{T}(G_1, S_1)$ and
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$\mathcal{T}_2 = \mathcal{T}(G_2, S_2)$ arising from maximal planar
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graphs $G_1, G_2$ with respective single-vertex level sources
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$S_1, S_2$, the following holds: $\mathcal{T}_1$ \emph{nests}
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into $\mathcal{T}_2$.
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By ``$\mathcal{T}_1$ nests into $\mathcal{T}_2$'' we mean:
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\begin{itemize}
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\item Choose any tire tread $T \in \mathcal{T}_2$ and any non-trivial
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bounded face $f$ of its inner outerplanar graph $O^{(T)}$
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(i.e.\ a face whose interior currently contains depth-$\ge d+2$
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vertices of $G_2$, where $d = \mathrm{depth}(T)$).
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\item Then there exists a maximal planar graph $\tilde G$ with
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level source $\tilde S$ such that:
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\begin{enumerate}
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\item[(N1)] $\mathcal{T}(\tilde G, \tilde S)$ contains
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$\mathcal{T}_2$ as a sub-tree (with every
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tire tread of $\mathcal{T}_2$ preserved
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combinatorially and embedded);
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\item[(N2)] the sub-tree of $\mathcal{T}(\tilde G, \tilde S)$
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rooted at the child of $T$ corresponding to face
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$f$ is isomorphic, as a rooted tree of tire treads,
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to $\mathcal{T}_1$.
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\end{enumerate}
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\end{itemize}
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\medskip
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Informally: any tree of tire treads can be ``inserted'' into any
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non-trivial face slot of any other tree of tire treads, producing
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a larger maximal planar graph whose tree of tire treads is the
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nested combination. The class of trees of tire treads is
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\emph{closed under composition} by face-slot insertion.
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\end{conjecture}
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\begin{remark}
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\label{rem:nesting-motivation}
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The conjectured closure under nesting carries two structural
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implications for the Four Colour Theorem programme:
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\begin{itemize}
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\item \emph{Compositional colourability.} If colourability of
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$\tilde G$ in (N1)--(N2) can be decided from the colourability
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of $G_1$ and $G_2$ alone (via the parent--child consistency
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constraints of Remark~\ref{rem:tree-coloring-factorisation}),
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then $4$-colourability propagates through nesting. A
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minimum $4$CT counterexample (if it exists) would have to be
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\emph{irreducible} under such nesting --- it could not be
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decomposed into strictly smaller trees of tire treads whose
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colourings combine to a colouring of the whole.
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\item \emph{Universality.} Universal nesting positions trees of
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tire treads as a kind of ``term algebra'' for the structural
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decomposition of plane triangulations. Coloring arguments
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can then be formulated inductively on this term algebra,
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with the chain-pigeonhole step
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(Remark~\ref{rem:tree-coloring-factorisation}) supplying the
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composition rule.
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\end{itemize}
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Open questions include: which precise notion of ``isomorphic as
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rooted trees of tire treads'' should be used (combinatorial,
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geometric, or up to embedding)? Does the nested triangulation
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$\tilde G$ admit a constructive description from $G_1, G_2$ and
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the choice of face $f$? And does nesting respect Birkhoff's
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internally $6$-connected condition for minimum counterexamples?
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\end{remark}
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\begin{thebibliography}{9}
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\begin{thebibliography}{9}
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