diff --git a/.codex/hooks.json b/.codex/hooks.json new file mode 100644 index 0000000..e7e3f63 --- /dev/null +++ b/.codex/hooks.json @@ -0,0 +1,38 @@ +{ + "hooks": { + "PostToolUse": [ + { + "hooks": [ + { + "type": "command", + "command": "python3 /Users/didericis/Code/voice-control/codex/hooks/speak-narration.py" + } + ] + } + ], + "Stop": [ + { + "hooks": [ + { + "type": "command", + "command": "/Users/didericis/Code/voice-control/codex/hooks/thinking-cue.sh stop" + }, + { + "type": "command", + "command": "python3 /Users/didericis/Code/voice-control/codex/hooks/speak-response.py" + } + ] + } + ], + "UserPromptSubmit": [ + { + "hooks": [ + { + "type": "command", + "command": "/Users/didericis/Code/voice-control/codex/hooks/thinking-cue.sh start" + } + ] + } + ] + } +} diff --git a/papers/coloring_nested_tire_graphs/fig_seam_construction.png b/papers/coloring_nested_tire_graphs/fig_seam_construction.png index a80da8e..4427322 100644 Binary files a/papers/coloring_nested_tire_graphs/fig_seam_construction.png and b/papers/coloring_nested_tire_graphs/fig_seam_construction.png differ diff --git a/papers/coloring_nested_tire_graphs/paper.aux b/papers/coloring_nested_tire_graphs/paper.aux index 50d2f16..9cd7bb5 100644 --- a/papers/coloring_nested_tire_graphs/paper.aux +++ b/papers/coloring_nested_tire_graphs/paper.aux @@ -29,15 +29,21 @@ \@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} 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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 $\Pi_G$