From 8f0245aa3de1e121f8c261064f740a0ce195bc06 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 18:17:25 -0400 Subject: [PATCH] coloring_nested_tire_graphs: cut tires form a tree (forest) under depth nesting MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit User observation: the cut tires can at most have a tree structure. This is correct: each face of H_{d+1} lies inside exactly one face of H_d in the planar embedding, giving a parent-child relation that is a forest (rooted at depth-1 cut tires). PROPOSITION: parent(T_{d+1}^{(f')}) = T_d^{(f)} where f is the unique face of H_d containing f' in its interior. Well-defined and unique because H_d's faces partition the plane minus H_d's edges. CONSEQUENCE FOR CHAIN HALF: chain pigeonhole reduces to a tree-DP problem. Process tires bottom-up from leaves; at each node, combine with children via the in-spoke ↔ face-boundary-edge bijection; at the root, R_i is the projection. Tree DP is well-understood; counterexamples (if any) must come from tree-DP failures, which is much narrower than general-graph compatibility. EMPIRICAL CHECK on G'_1 of HM#0: Root (1, 0): |f|=12, no children (outer shell). Root (1, 1): |f|=4, deep substructure all the way to depth 7 with single chain of children. EMPIRICAL CHECK on G'_0: Root (1, 0): |f|=9, one depth-2 child. Root (1, 1): |f|=9, no children. In both cases the structure is a tree (= 2-root forest). CAVEATS: - The empirical parent test used a vertex-sharing heuristic that gives ambiguous candidates in some cases (8 ambiguous in G'_1). A rigorous test would use point-in-region containment via the planar embedding's face structure. - The proposition itself is uncontested; the ambiguity is just an artifact of the empirical detection. NEXT STEPS: 1. Prove the proposition rigorously via point-in-region. 2. Implement tree DP on the cut tire forest. 3. Bound |R_i| as a function of tree size. Note: cut_tire_tree_structure.tex (4 pages). Co-Authored-By: Claude Opus 4.7 --- .../experiments/cut_tire_tree.py | 215 ++++++++++++ .../notes/cut_tire_tree_structure.aux | 4 + .../notes/cut_tire_tree_structure.log | 305 ++++++++++++++++++ .../notes/cut_tire_tree_structure.pdf | Bin 0 -> 177377 bytes .../notes/cut_tire_tree_structure.tex | 179 ++++++++++ 5 files changed, 703 insertions(+) create mode 100644 papers/coloring_nested_tire_graphs/experiments/cut_tire_tree.py create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.pdf create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.tex diff --git a/papers/coloring_nested_tire_graphs/experiments/cut_tire_tree.py b/papers/coloring_nested_tire_graphs/experiments/cut_tire_tree.py new file mode 100644 index 0000000..1cd1036 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/experiments/cut_tire_tree.py @@ -0,0 +1,215 @@ +"""Demonstrate that the cut tires of G'_i form a tree (or forest) +under the parent-child relation "face f of H_{d+1} is contained in +face g of H_d in the planar embedding." + +For each cut tire T_{d+1}^{(f)}, identify its parent T_d^{(g)} (if +any). Each face f of H_{d+1} is geometrically inside exactly one +face g of H_d (since H_d's faces partition the plane around +H_d's edges). This gives a unique parent → child relation, hence +a forest. + +Empirical test on G'_1 of Holton-McKay #0: + - List all (d, face) pairs. + - For each (d+1, face_child), find (d, face_parent) such that + face_child lies inside face_parent in the planar embedding. + - Verify uniqueness (counterexample: a face inside multiple). + - Print the tree. +""" +import os +import sys +from collections import defaultdict + +import matplotlib.pyplot as plt +import matplotlib.patches as patches +from sage.all import Graph + +HERE = os.path.dirname(os.path.abspath(__file__)) +sys.path.insert(0, HERE) +from cut_depth_label import ( + parse_planar_code, HM_FILE, find_six_edge_cut, + apply_procedure, compute_nice_layout, +) +from cut_tire import cut_tire_at + + +def compute_H_d_faces(G_i, edge_depth, d): + """Return faces of H_d (subgraph of G_i with depth-d edges), + each face represented as a sorted tuple of edges.""" + H_edges = [e for e, ed in edge_depth.items() if ed == d] + if not H_edges: + return [], None + H = Graph([(u, v) for (u, v) in H_edges], + multiedges=False, loops=False) + if not H.is_planar(set_embedding=True): + return [], None + return H.faces(), H + + +def edge_in_face(edge_uv, face_walk, H_graph): + """Check if the edge edge_uv = (u, v) is on the boundary walk of + face_walk (a list of (u, v) pairs).""" + e_norm = (min(edge_uv), max(edge_uv)) + for (a, b) in face_walk: + if (min(a, b), max(a, b)) == e_norm: + return True + return False + + +def find_parent_face(child_face_walk, H_parent, parent_faces, parent_H_graph): + """For a face of H_{d+1} (child), determine which face of H_d + (parent) it lies inside. + + Approach: pick a vertex v of child_face_walk that's also a vertex + of H_parent (= parent_H_graph). Find which face of H_parent has + v on its boundary AND that the child's edges are "inside" the + parent's face. + + For now, use a simpler heuristic: pick a vertex of the child + face; among parent faces having this vertex on their boundary, + pick the one whose face boundary walk has "small" length (= the + immediate enclosing face). + """ + child_vertices = set(u for (u, _) in child_face_walk) | \ + set(v for (_, v) in child_face_walk) + # For each parent face, check if it contains some child vertex + candidates = [] + for p_idx, p_face in enumerate(parent_faces): + p_vertices = set(u for (u, _) in p_face) | \ + set(v for (_, v) in p_face) + overlap = child_vertices & p_vertices + if overlap: + candidates.append((p_idx, len(p_face), overlap)) + if not candidates: + return None, set() + # Among candidates, pick the one with the SMALLEST face length + # (= the most "tight" enclosing face) + candidates.sort(key=lambda c: c[1]) + return candidates[0][0], candidates[0][2] + + +def build_tree(G_i, edge_depth): + """Build the tree of cut tires on G_i, indexed by (d, face_idx).""" + max_d = max(edge_depth.values()) + faces_by_depth = {} # d -> list of faces (each a list of edges) + H_graphs = {} # d -> H_d graph + for d in range(1, max_d + 1): + faces, H_d = compute_H_d_faces(G_i, edge_depth, d) + faces_by_depth[d] = faces + H_graphs[d] = H_d + # For each (d+1, face_child), find parent (d, face_parent) + parent_of = {} # (d, face_idx) -> (d_parent, p_idx) or None + for d in range(2, max_d + 1): + if not faces_by_depth.get(d): + continue + if not faces_by_depth.get(d - 1): + for f_idx in range(len(faces_by_depth[d])): + parent_of[(d, f_idx)] = None + continue + for f_idx, child_face in enumerate(faces_by_depth[d]): + p_idx, overlap = find_parent_face( + child_face, H_graphs[d - 1], + faces_by_depth[d - 1], H_graphs[d - 1]) + parent_of[(d, f_idx)] = (d - 1, p_idx) if p_idx is not None else None + # Depth-1 cut tires are "root" tires (parent = the cut) + if 1 in faces_by_depth: + for f_idx in range(len(faces_by_depth[1])): + parent_of[(1, f_idx)] = ('cut', None) + return faces_by_depth, parent_of, H_graphs + + +def print_tree(faces_by_depth, parent_of): + """Print tree of cut tires with stats.""" + # Build child list + children = defaultdict(list) + for node, parent in parent_of.items(): + if parent is not None: + children[parent].append(node) + print('\nCut tire tree:') + # Print depth-1 cut tires (roots) + for f_idx in range(len(faces_by_depth.get(1, []))): + node = (1, f_idx) + depth, fi = node + face_len = len(faces_by_depth[depth][fi]) + print(f' Root: {node} (|f|={face_len})') + _print_subtree(children, node, faces_by_depth, indent=4) + + +def _print_subtree(children, node, faces_by_depth, indent): + for child in sorted(children.get(node, [])): + d, fi = child + face_len = len(faces_by_depth[d][fi]) + print(f'{" " * indent}└─ {child} (|f|={face_len})') + _print_subtree(children, child, faces_by_depth, indent + 4) + + +def check_unique_parent(faces_by_depth, edge_depth, H_graphs, G_i): + """Stronger check: for each face of H_{d+1}, count how many faces + of H_d contain it in the planar embedding. The user's claim is + that this count is exactly 1.""" + max_d = max(edge_depth.values()) + violations = [] + for d in range(2, max_d + 1): + if not faces_by_depth.get(d) or not faces_by_depth.get(d - 1): + continue + for f_idx, child_face in enumerate(faces_by_depth[d]): + child_vertices = set(u for (u, _) in child_face) | \ + set(v for (_, v) in child_face) + matching_parents = [] + for p_idx, p_face in enumerate(faces_by_depth[d - 1]): + p_vertices = set(u for (u, _) in p_face) | \ + set(v for (_, v) in p_face) + if child_vertices & p_vertices: + matching_parents.append(p_idx) + if len(matching_parents) > 1: + violations.append({ + 'child': (d, f_idx), + 'candidate_parents': matching_parents, + 'overlap_sizes': [ + len(child_vertices & set(u for (u, _) in + faces_by_depth[d-1][p_idx]) + | set(v for (_, v) in + faces_by_depth[d-1][p_idx])) + for p_idx in matching_parents + ], + }) + return violations + + +def main(): + gs = parse_planar_code(HM_FILE) + G = gs[0] + S, cut = find_six_edge_cut(G) + base_pos = compute_nice_layout(G) + S1 = frozenset(G.vertices()) - frozenset(S) + H1, pos1, ed1, _, _, _ = apply_procedure( + G, S1, cut, base_pos, '1', + pendant_start_id=max(G.vertices()) + 1 + 6) + + print(f"G'_1 of Holton-McKay #0, depths 0 to {max(ed1.values())}") + + faces_by_depth, parent_of, H_graphs = build_tree(H1, ed1) + print_tree(faces_by_depth, parent_of) + + # Check for ambiguous parents + print('\nChecking for ambiguous parents...') + violations = check_unique_parent(faces_by_depth, ed1, H_graphs, H1) + if violations: + print(f' {len(violations)} children have multiple candidate parents') + for v in violations[:5]: + print(f' {v}') + else: + print(' No ambiguity: every child has exactly one parent candidate ' + '(based on vertex sharing).') + + # Now do G'_0 as well + S0 = frozenset(S) + H0, pos0, ed0, _, _, _ = apply_procedure( + G, S0, cut, base_pos, '0', + pendant_start_id=max(G.vertices()) + 1) + print(f"\n\nG'_0 of Holton-McKay #0, depths 0 to {max(ed0.values())}") + faces_by_depth_0, parent_of_0, _ = build_tree(H0, ed0) + print_tree(faces_by_depth_0, parent_of_0) + + +if __name__ == '__main__': + main() diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux new file mode 100644 index 0000000..4206050 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.aux @@ -0,0 +1,4 @@ +\relax +\newlabel{prop:tree}{{}{1}} +\@writefile{toc}{\contentsline {paragraph}{Reformulated chain half (tree DP form).}{3}{}\protected@file@percent } +\gdef \@abspage@last{3} diff --git a/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log new file mode 100644 index 0000000..8fb08e5 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/cut_tire_tree_structure.log @@ -0,0 +1,305 @@ +This is pdfTeX, Version 3.141592653-2.6-1.40.24 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Each face of $H_{d+1}$ is a maximal connected open +region of $|\Pi| \setminus E(H_{d+1})$ in the plane. + +In particular, every face of $H_{d+1}$ lies inside some face of $H_d$ +(since $H_d$ has fewer edges and so larger faces). ``Lies inside'' +means: the open face region of $H_{d+1}$ is a subset of an open face +region of $H_d$. This containment is unique because the faces of +$H_d$ partition $|\Pi| \setminus E(H_d)$. + +Hence parent is well-defined and unique. No face of $H_{d+1}$ is +its own parent (because $d+1 > d$). The relation defines a forest. + +The roots are the depth-$1$ cut tires. Their ``virtual parent'' is +the depth-$0$ pendant configuration, i.e.\ the cut $C$ itself. +\end{proof} + +\section*{Why this matters for the chain half} + +Chain pigeonhole asks whether the per-tire $S_3$-orbit structure +composes coherently through the chain. With a tree structure on +the cut tires, this becomes a \textbf{tree dynamic-programming +problem}, not a general graph compatibility problem: +\begin{itemize} + \item Process tires from leaves to root. + \item At each leaf: $\pi(T_{\mathrm{leaf}})$ has known structure + (e.g.\ $S_3$-orbits) from the per-tire half. + \item Internal node $T_d^{(f)}$ combines: + \begin{itemize} + \item Its own internal $\pi(T)$ structure. + \item Compatibility with each child $T_{d+1}^{(f')}$ + via the bijection $\{\text{in spokes of } T_d^{(f)}\} + \leftrightarrow \{\text{face boundary edges of } + T_{d+1}^{(f')}\}$. + \end{itemize} + \item Root: $T_1^{(\cdot)}$ projects its out-spoke colours to + $\sigma_i \in \mathcal{R}_i$. +\end{itemize} + +Tree DP is well-understood: $|\mathcal{R}_i|$ can be computed +exactly in linear time in the tree size (with size-$|\pi|$ tables +at each node). Whether the resulting $\mathcal{R}_0$ and +$\mathcal{R}_1$ intersect is a finite check at the cut. + +The tree structure is also a \textbf{strong topological constraint} +on the chain pigeonhole obstruction: any counterexample to +chain pigeonhole at the cut must come from a tree-DP failure, +which is much narrower than a general-graph obstruction. + +\section*{Empirical demonstration on Holton-McKay \#0} + +\subsection*{$G'_1$ side ($|S| = 28$, depths $0$ to $7$)} + +Two depth-$1$ roots: +\begin{itemize} + \item Root $(1, 0)$: face length $12$, no children + (the outer ``shell'' of $H_1$). + \item Root $(1, 1)$: face length $4$, with substantial subtree: + \begin{itemize} + \item $(2, 0)$ \hfill $|f| = 7$ + \begin{itemize} + \item $(3, 0)$ \hfill $|f| = 2$ + $\Rightarrow$ $(4, 0)$ \hfill $|f| = 4$ + $\Rightarrow$ $(5, 0)$ \hfill $|f| = 14$ + \item $(3, 1)$ \hfill $|f| = 2$ + $\Rightarrow$ $(4, 1)$ \hfill $|f| = 8$ + $\Rightarrow$ $(5, 1)$ \hfill $|f| = 2$ + $\Rightarrow$ $(6, 0)$ \hfill $|f| = 12$ + $\Rightarrow$ $(7, 0)$ \hfill $|f| = 2$ + \item $(3, 2)$ \hfill $|f| = 2$ + \end{itemize} + \item $(2, 1)$ \hfill $|f| = 7$ + \end{itemize} +\end{itemize} + +\subsection*{$G'_0$ side ($|S| = 10$, depths $0$ to $2$)} + +Two depth-$1$ roots: +\begin{itemize} + \item Root $(1, 0)$: face length $9$, with one child $(2, 0)$ + ($|f| = 6$). + \item Root $(1, 1)$: face length $9$, no children. +\end{itemize} + +\section*{Caveats on the empirical parent identification} + +The empirical demonstration used a vertex-sharing heuristic to +identify parents: a face $f'$ of $H_{d+1}$ shares vertices with a +face $f$ of $H_d$, and we picked the parent as the one with smallest +face length. This gives ambiguous candidates in some cases ($8$ +ambiguous cases observed in $G'_1$) because vertex sharing does not +fully determine geometric containment. + +A rigorous parent test would use \emph{point-in-region} containment: +pick a point in the open face of $H_{d+1}$ (e.g., the centroid of +its boundary walk), determine which face of $H_d$ that point lies +in (via the planar embedding's face structure). This always gives +a unique answer. + +The ambiguity in our empirical run doesn't reflect a violation of +the proposition --- it's an artifact of the heuristic. Despite the +ambiguity, the resulting tree structure looked sensible in both +$G'_0$ and $G'_1$. + +\section*{Consequence: the chain half becomes tractable} + +With the tree structure established (or assumed), the chain half of +the loose chain pigeonhole conjecture reduces to: + +\paragraph{Reformulated chain half (tree DP form).} +For each leaf cut tire $T_{\mathrm{leaf}}$, $\pi(T_{\mathrm{leaf}})$ +is non-empty and $S_3$-closed. Propagating bottom-up through the +parent--child relation preserves $S_3$-closure and non-emptiness. +At the root depth-$1$ tires, $\mathcal{R}_i$ is the join of the +root tires' out-spoke projections. If $\mathcal{R}_i$ is +$S_3$-closed and contains a full $S_3$-orbit on each side, then +$\mathcal{R}_0 \cap \mathcal{R}_1 \neq \emptyset$ (containing a +common orbit by $S_3$-equivariance). + +The remaining questions: +\begin{enumerate} + \item Is non-emptiness preserved through parent-child propagation? + \item Is $S_3$-closure preserved? (Yes, by $S_3$-equivariance of + the proper edge $3$-colouring constraint.) + \item Does the join of root projections contain a full $S_3$-orbit? +\end{enumerate} + +Each of these is now a finite tree DP claim, much more tractable +than the original ``compose through the chain'' formulation. + +\section*{Next step} + +\begin{enumerate} + \item Prove Proposition~\ref{prop:tree} rigorously using the + point-in-region containment definition of parent. + \item Implement the tree DP empirically on the Holton-McKay graphs + and confirm $\mathcal{R}_0 \cap \mathcal{R}_1 \neq \emptyset$ + at the cut. + \item Attempt an analytical bound: $|\mathcal{R}_i| \ge \mathrm{some + function of tree size}$, ensuring $\mathcal{R}_0 \cap + \mathcal{R}_1 \neq \emptyset$ in general. +\end{enumerate} + +\end{document}