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Add tire chromatic transfer content

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didericis
2026-06-01 01:43:06 -04:00
parent 236a3f8288
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8 changed files with 848 additions and 52 deletions
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papers/coloring_nested_tire_graphs/fig_seam_construction.png
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papers/coloring_nested_tire_graphs/paper.aux
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\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Case 2 ($R$ = annulus) with $O$ a barbell. $B_{\mathrm {out}}$ is the outer hexagon (red); $O$ has two triangles $\{a_1, a_2, a_3\}$ and $\{b_1, b_2, b_3\}$ joined by the bridge $a_3\text {--}b_1$ (all light red). The annulus is triangulated by $14$ annular triangles: $6$ ``outer-cap'' triangles (one per outer edge), $6$ ``inner-cap'' triangles (one per non-bridge edge of $O$), and $2$ ``bridge-cap'' triangles $\{u_0, a_3, b_1\}$ and $\{u_3, a_3, b_1\}$ adjacent to the bridge. Each blue dot sits at the centroid of an annular triangle; blue edges connect dual vertices whose triangles share an interior annular edge (spoke or bridge). The two bridge-cap vertices have $\Gamma $-degree $3$ (their triangles have no boundary edge) and are joined by the dashed blue \emph {chord} corresponding to the bridge; the remaining $13$ edges form the Hamilton cycle that wraps around the annulus. All $14$ vertices lie on the outer face of the cycle-with-chord embedding, so $\Gamma \cong \Theta (1, 7, 7)$ is outerplanar.}}{10}{}\protected@file@percent }
\newlabel{fig:inner-dual-annulus-case}{{4}{10}}
\newlabel{rem:count-general-outerplanar}{{1.17}{11}}
\newlabel{thm:tread-tree}{{1.18}{11}}
\newlabel{rem:tree-multiple-children}{{1.19}{12}}
\newlabel{thm:tire-tree-decomposition}{{1.20}{12}}
\newlabel{rem:tree-coloring-factorisation}{{1.21}{14}}
\newlabel{def:seam}{{1.22}{14}}
\newlabel{def:partial-tire-tree}{{1.23}{14}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.20\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{15}{}\protected@file@percent }
\newlabel{fig:tire-tree-decomposition}{{5}{15}}
\newlabel{lem:seam-edge-shared}{{1.24}{15}}
\newlabel{def:boundary-state-transfer}{{1.18}{11}}
\newlabel{thm:tire-chromatic-polynomial-transfer}{{1.19}{11}}
\newlabel{rem:spoke-only-chromatic-transfer}{{1.20}{12}}
\newlabel{thm:tread-tree}{{1.21}{12}}
\newlabel{rem:tree-multiple-children}{{1.22}{13}}
\newlabel{thm:tire-tree-decomposition}{{1.23}{13}}
\newlabel{rem:tree-coloring-factorisation}{{1.24}{15}}
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces Tire-tree decomposition (Theorem\nonbreakingspace 1.23\hbox {}) on a $13$-vertex maximal planar example $G$ with five BFS levels. $(a)$ $G$ with vertex source $v_0$ and $\ell _G \in \{0,1,2,3,4\}$; four nested seams are highlighted, $C_{T_R} = \{a,b,c\}$ (orange), $C_{T_L} = \{a,c,d\}$ (red, including the chord $a$-$c$ shared with $C_{T_R}$), $C_{T_{LL}} = \{f_1, f_2, f_3\}$ (purple), $C_{T_{LLL}} = \{g_1, g_2, g_3\}$ (teal). Inset: the rooted tree of tire treads $\mathcal {T}(G, \{v_0\})$ branches at $T_0$ into the leaf $T_R$ (containing $e$) and a chain $T_L \to T_{LL} \to T_{LLL}$ (the highlighted sub-tree). $(b)$ The disk $G_{T_L}$ inside the seam $C_{T_L}$, drawn standalone with $C_{T_L}$ as cycle source and vertex labels rotated to match the new (cycle-source) role of the boundary triangle. $\ell _{G_{T_L}}(\cdot ) = \ell _G(\cdot ) - 1$ on $V(G_{T_L})$ (verified by the generator script), and $\mathcal {T}(G_{T_L}, C_{T_L})$ is the chain $T_L \to T_{LL} \to T_{LLL}$, iso to the highlighted sub-tree of $(a)$.}}{16}{}\protected@file@percent }
\newlabel{fig:tire-tree-decomposition}{{5}{16}}
\newlabel{def:level-cycle-three-colour-restriction}{{1.25}{16}}
\newlabel{conj:level-cycle-three-colour}{{1.26}{16}}
\newlabel{def:seam}{{1.27}{16}}
\newlabel{def:partial-tire-tree}{{1.28}{17}}
\newlabel{lem:seam-edge-shared}{{1.29}{17}}
\newlabel{conj:seam-counterexample}{{1.30}{17}}
\bibcite{tait-original}{1}
\bibcite{bauerfeld-depth}{2}
\bibcite{bauerfeld-nested-tire-duals}{3}
@@ -46,6 +52,5 @@
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\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\newlabel{conj:seam-counterexample}{{1.25}{16}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{16}{}\protected@file@percent }
\gdef \@abspage@last{16}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{18}{}\protected@file@percent }
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papers/coloring_nested_tire_graphs/paper.fdb_latexmk
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@@ -861,6 +861,107 @@ the edge-chromatic polynomial admits a deletion--contraction
recursion that respects the cycle-plus-chord structure.
\end{remark}
\begin{definition}[Boundary-state chromatic transfer]
\label{def:boundary-state-transfer}
Let $T = (B_{\mathrm{out}}, O, E_{\mathrm{ann}})$ be a tire graph.
Choose a cut along one annular edge if both boundaries are
non-degenerate; in the degenerate case make no cut. The tread becomes
a triangulated disk $\widetilde R$. Let
\[
f_1, f_2, \ldots, f_m
\]
be any shelling order of the triangular faces of $\widetilde R$, i.e.\
an order in which each initial union
$\widetilde R_i := f_1 \cup \cdots \cup f_i$ is a disk. Such an order
is obtained by taking an outerplanar embedding of the inner dual
$\Gamma$ from Theorem~\ref{thm:inner-dual-outerplanar} and repeatedly
removing an outer-face ear.
For each $i$, let $A_i$ be the \emph{frontier}: the vertices of
$T$ incident to at least one processed face in $\widetilde R_i$ and to
at least one still-unprocessed constraint, where the unprocessed
constraints are the remaining annular faces together with any edge of
$O$ not yet tested by the transfer. A \emph{boundary state} on $A_i$
is a partition $\pi$ of $A_i$ into colour classes, subject to the
condition that adjacent vertices of $T[A_i]$ lie in distinct blocks.
We write $r(\pi)$ for the number of blocks of $\pi$.
\end{definition}
\begin{theorem}[Chromatic polynomial of a tire by frontier transfer]
\label{thm:tire-chromatic-polynomial-transfer}
For every tire graph $T$, the chromatic polynomial $P_T(q)$ is
computed by the following boundary-state dynamic program.
Initialize the table at $i=0$ with the empty frontier state of weight
$1$. When the next triangular face
$f_i = \{x,y,z\}$ is attached, pass from states on $A_{i-1}$ to states
on $A_i$ as follows.
\begin{enumerate}
\item[(1)] Introduce any vertices of $f_i$ not already present in
$A_{i-1}$, assigning each such vertex either to an existing
colour block not containing one of its already-coloured
neighbours, or to a new block.
\item[(2)] Reject every assignment in which two adjacent vertices of
the triangle $f_i$ lie in the same block. Also reject every
assignment in which an edge of $O$ whose two endpoints have
now both appeared for the first time as a tested pair has
both endpoints in the same block. Thus chords and bridges of
the inner outerplanar graph are enforced exactly when their
second endpoint becomes visible to the transfer.
\item[(3)] Delete from the state every vertex no longer incident to an
unprocessed constraint. If deleting a vertex removes the last
representative of its colour block from the frontier, multiply
that transition by $1$; the colour has already been chosen.
\item[(4)] If a new vertex is assigned to a new colour block while the
current frontier state has $r$ colour blocks, multiply that
transition by $q-r$. If several new colour blocks are created
in the same triangle, the factors are
$(q-r)(q-r-1)\cdots$ in the order of creation.
\end{enumerate}
After $f_m$ is processed, the frontier is empty. The single remaining
weight is $P_T(q)$.
\end{theorem}
\begin{proof}
The construction is the standard transfer for the chromatic polynomial,
specialized to the tire shelling. The frontier state records exactly
the equality pattern among colours that can still affect unprocessed
faces. Since colour names are irrelevant to the chromatic polynomial,
states are quotiented by the natural action of the symmetric group on
the colour set; a state with $r$ visible colour blocks can be extended
by a genuinely new colour in $q-r$ ways.
Each transition accounts for all proper colourings of the enlarged
processed disk $\widetilde R_i$ that restrict to the resulting frontier
state, and accounts for none that violate an edge of the newly attached
triangle or untested edge of $O$. Vertices removed from the frontier
have no future incident unprocessed constraint, so their actual colour
names can no longer influence compatibility and may be forgotten.
Induction on $i$ therefore shows
that the table after step $i$ is precisely the orbit-count generating
function for proper colourings of $\widetilde R_i$ by frontier state.
At $i=m$ no vertices remain active, so the accumulated weight counts all
proper colourings of $T$. Because the weights are polynomials in $q$,
this count is the full chromatic polynomial.
\end{proof}
\begin{remark}[Spoke-only transfer matrix]
\label{rem:spoke-only-chromatic-transfer}
In the spoke-only case with both boundaries simple cycles, the method
has a particularly small form. Cut the annulus along one spoke and
walk around the resulting strip. Each step adds one triangle sharing
an edge with the previous processed strip, so the frontier consists of
two or three consecutive boundary vertices. Up to colour permutation
there are only the possible equality patterns among those active
vertices, with adjacent vertices required to be distinct. The
chromatic polynomial is therefore the trace of a finite transfer matrix
whose entries are polynomials in $q$; the matrix depends only on the
local triangle type encountered while walking around the tread. Chords
or cut-vertices of $O$ enlarge the frontier only at the corresponding
outerplanar ears, and are handled by the same state rule of
Theorem~\ref{thm:tire-chromatic-polynomial-transfer}.
\end{remark}
\begin{theorem}[Tire treads form a rooted tree under face containment]
\label{thm:tread-tree}
Let $G$ be a maximal planar graph with planar embedding $\Pi_G$