Add tire chromatic transfer content
This commit is contained in:
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{
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"hooks": {
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"PostToolUse": [
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{
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"hooks": [
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{
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"type": "command",
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"command": "python3 /Users/didericis/Code/voice-control/codex/hooks/speak-narration.py"
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}
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]
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}
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],
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"Stop": [
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{
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"hooks": [
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{
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"type": "command",
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"command": "/Users/didericis/Code/voice-control/codex/hooks/thinking-cue.sh stop"
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},
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{
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"type": "command",
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"command": "python3 /Users/didericis/Code/voice-control/codex/hooks/speak-response.py"
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}
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]
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}
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],
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"UserPromptSubmit": [
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{
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"hooks": [
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{
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"type": "command",
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"command": "/Users/didericis/Code/voice-control/codex/hooks/thinking-cue.sh start"
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}
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]
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}
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]
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}
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}
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@@ -29,15 +29,21 @@
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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 }
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\newlabel{fig:inner-dual-annulus-case}{{4}{10}}
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\newlabel{rem:count-general-outerplanar}{{1.17}{11}}
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\newlabel{thm:tread-tree}{{1.18}{11}}
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\newlabel{rem:tree-multiple-children}{{1.19}{12}}
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\newlabel{thm:tire-tree-decomposition}{{1.20}{12}}
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\newlabel{rem:tree-coloring-factorisation}{{1.21}{14}}
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\newlabel{def:seam}{{1.22}{14}}
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\newlabel{def:partial-tire-tree}{{1.23}{14}}
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\@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 }
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\newlabel{fig:tire-tree-decomposition}{{5}{15}}
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\newlabel{lem:seam-edge-shared}{{1.24}{15}}
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\newlabel{def:boundary-state-transfer}{{1.18}{11}}
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\newlabel{thm:tire-chromatic-polynomial-transfer}{{1.19}{11}}
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\newlabel{rem:spoke-only-chromatic-transfer}{{1.20}{12}}
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\newlabel{thm:tread-tree}{{1.21}{12}}
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\newlabel{rem:tree-multiple-children}{{1.22}{13}}
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\newlabel{thm:tire-tree-decomposition}{{1.23}{13}}
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\newlabel{rem:tree-coloring-factorisation}{{1.24}{15}}
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\@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 }
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\newlabel{fig:tire-tree-decomposition}{{5}{16}}
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\newlabel{def:level-cycle-three-colour-restriction}{{1.25}{16}}
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\newlabel{conj:level-cycle-three-colour}{{1.26}{16}}
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\newlabel{def:seam}{{1.27}{16}}
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\newlabel{def:partial-tire-tree}{{1.28}{17}}
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\newlabel{lem:seam-edge-shared}{{1.29}{17}}
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\newlabel{conj:seam-counterexample}{{1.30}{17}}
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\bibcite{tait-original}{1}
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\bibcite{bauerfeld-depth}{2}
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\bibcite{bauerfeld-nested-tire-duals}{3}
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@@ -46,6 +52,5 @@
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\newlabel{tocindent1}{17.77782pt}
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\newlabel{tocindent2}{0pt}
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\newlabel{tocindent3}{0pt}
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\newlabel{conj:seam-counterexample}{{1.25}{16}}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{16}{}\protected@file@percent }
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\gdef \@abspage@last{16}
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\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{18}{}\protected@file@percent }
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\gdef \@abspage@last{18}
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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 $\Pi_G$
|
||||
|
||||
Reference in New Issue
Block a user