From cb6a79f799484490f562acc13de923403f3143b4 Mon Sep 17 00:00:00 2001 From: didericis Date: Tue, 26 May 2026 15:56:26 -0400 Subject: [PATCH] coloring_nested_tire_graphs: redefine cut tire bullet 2 as labelled pendants MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Per user spec: instead of including the actual depth-(d±1) edges incident to the face boundary, redefine the cut tire as: - Face boundary walk of f (depth-d edges in H_d). - For each vertex v on the boundary walk with degree-2 in H_d: add a fresh vertex n_v and fresh edge {v, n_v}, labelled "out spoke" if v has an incident depth-(d-1) edge in G'_i, "in spoke" if v has an incident depth-(d+1) edge. Result: each cut tire is intrinsically "cycle (or closed walk) + labelled pendants," structurally isomorphic to the partial tire dual D(T) from paper.tex. Pendants ↔ D(T)'s leaves, face boundary ↔ T'_ann. This means propositions about D(T) (chromatic polynomial counts, S_3-orbit structure, rainbow conjecture, etc.) apply verbatim to each cut tire. Updates: - notes/cut_depth_label.tex: Definition rewritten, structural remark added, table of spoke counts updated to match new defn. - experiments/cut_tire.py: cut_tire_at() now computes labelled pendants instead of incident edges; draw_cut_tire renders pendant vertices (orange squares for out, green squares for in) with edges offset toward parent-graph neighbor. - notes/fig_cut_tire.png: regenerated. Note grows to 6 pages. Co-Authored-By: Claude Opus 4.7 --- .../experiments/cut_tire.py | 205 +++++++---- .../notes/cut_depth_label.aux | 10 +- .../notes/cut_depth_label.fdb_latexmk | 84 +++++ .../notes/cut_depth_label.fls | 343 ++++++++++++++++++ .../notes/cut_depth_label.log | 42 ++- .../notes/cut_depth_label.pdf | Bin 504231 -> 509609 bytes .../notes/cut_depth_label.tex | 87 +++-- .../notes/fig_cut_tire.png | Bin 227925 -> 232836 bytes 8 files changed, 637 insertions(+), 134 deletions(-) create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_depth_label.fdb_latexmk create mode 100644 papers/coloring_nested_tire_graphs/notes/cut_depth_label.fls diff --git a/papers/coloring_nested_tire_graphs/experiments/cut_tire.py b/papers/coloring_nested_tire_graphs/experiments/cut_tire.py index 3b339fd..fb93061 100644 --- a/papers/coloring_nested_tire_graphs/experiments/cut_tire.py +++ b/papers/coloring_nested_tire_graphs/experiments/cut_tire.py @@ -40,13 +40,15 @@ def faces_of_subgraph(H, pos): def cut_tire_at(G_i, edge_depth, d): - """Build cut tires at depth d for each face of the depth-d subgraph. - Returns list of (face_vertices, tire_subgraph_edges_by_role) where: - tire_subgraph_edges_by_role is a dict with keys - 'face' (boundary at depth d), 'inner' (adjacent at d-1), - 'outer' (adjacent at d+1). + """Build cut tires at depth d for each face of H_d. + + Per the (redefined) Definition, each cut tire is: + - The face boundary walk of f (edges of depth d in H_d), AND + - For each vertex v on the boundary walk with degree-2 in H_d: + a single pendant (fresh vertex n_v + fresh edge {v, n_v}) + labelled 'out spoke' if v has an incident depth-(d-1) edge + in G'_i, or 'in spoke' if v has an incident depth-(d+1) edge. """ - # Build H_d H_d_edges = [e for e, ed in edge_depth.items() if ed == d] H_d = Graph([(u, v) for (u, v) in H_d_edges], multiedges=False, loops=False) @@ -56,68 +58,124 @@ def cut_tire_at(G_i, edge_depth, d): return [] faces_h_d = H_d.faces() results = [] + # For each vertex in H_d, compute its degree in H_d and find its + # incident non-H_d edges in G'_i (with their depths). + h_d_degree = {v: H_d.degree(v) for v in H_d.vertices()} + incident_other = {v: [] for v in H_d.vertices()} + for e, ed in edge_depth.items(): + if ed == d: + continue + u, w = e + if u in incident_other: + incident_other[u].append((w, ed)) + if w in incident_other: + incident_other[w].append((u, ed)) for face in faces_h_d: - # face is a list of (u, v) edge-pairs going around the boundary - # in Sage; convert to vertex sequence - vertices_on_face = [] - for (u, v) in face: - vertices_on_face.append(u) - face_vertex_set = set(vertices_on_face) - # Boundary edges of the face (in H_d): they're the face edges + vertices_on_face = [u for (u, v) in face] + face_vertex_list = list(dict.fromkeys(vertices_on_face)) # ordered, unique face_edges = [(min(u, v), max(u, v)) for (u, v) in face] - # Find adjacent edges at d-1 and d+1 from G'_i - inner_edges = [] # depth d-1 - outer_edges = [] # depth d+1 - for e_full, ed in edge_depth.items(): - if ed not in (d - 1, d + 1): + # For each degree-2 boundary vertex, add a labelled pendant + out_spokes = [] # list of (boundary_v, pendant_v) + in_spokes = [] # list of (boundary_v, pendant_v) + pendant_to_boundary = {} + for v in face_vertex_list: + if h_d_degree[v] != 2: continue - u, v = e_full - if u in face_vertex_set or v in face_vertex_set: + # In a cubic G'_i, v has exactly one non-H_d edge. + others = incident_other.get(v, []) + if not others: + continue + # Look at the depth of the unique non-H_d incident edge. + for (w, ed) in others: if ed == d - 1: - inner_edges.append(e_full) - else: - outer_edges.append(e_full) + pendant_id = ('out', v) + pendant_to_boundary[pendant_id] = v + out_spokes.append((v, pendant_id)) + elif ed == d + 1: + pendant_id = ('in', v) + pendant_to_boundary[pendant_id] = v + in_spokes.append((v, pendant_id)) results.append({ - 'face_vertices': vertices_on_face, + 'face_vertices_unique': face_vertex_list, 'face_edges': face_edges, - 'inner_edges': inner_edges, - 'outer_edges': outer_edges, + 'out_spokes': out_spokes, # (boundary_v, pendant_id) for d-1 + 'in_spokes': in_spokes, # (boundary_v, pendant_id) for d+1 'face_length': len(face), }) return results def draw_cut_tire(ax, G_i, pos, edge_depth, tire, d, title): - """Draw G_i faded with a single cut tire highlighted.""" + """Draw G_i faded with a single cut tire highlighted. + + The cut tire (under the redefinition) is: + - face boundary edges (depth d in H_d), and + - pendant edges from degree-2 boundary vertices: 'out spokes' + for those incident to a depth-(d-1) edge in G'_i, 'in spokes' + for those incident to a depth-(d+1) edge. + """ face_set = set(tire['face_edges']) - inner_set = set(tire['inner_edges']) - outer_set = set(tire['outer_edges']) - face_vertex_set = set(tire['face_vertices']) + face_vertex_set = set(tire['face_vertices_unique']) + out_spokes = tire['out_spokes'] + in_spokes = tire['in_spokes'] # Fade all G_i edges for e in G_i.edges(labels=False): u, v = e e_norm = (min(u, v), max(u, v)) - if e_norm in face_set or e_norm in inner_set or e_norm in outer_set: + if e_norm in face_set: continue x1, y1 = pos[u]; x2, y2 = pos[v] ax.plot([x1, x2], [y1, y2], color='#dddddd', linewidth=0.9, zorder=1) - # Highlight the tire + # Highlight the face boundary for e in face_set: u, v = e x1, y1 = pos[u]; x2, y2 = pos[v] ax.plot([x1, x2], [y1, y2], color='#1f77b4', linewidth=3.0, - zorder=3, label=None) - for e in inner_set: - u, v = e - x1, y1 = pos[u]; x2, y2 = pos[v] + zorder=3) + + # Compute pendant positions: each pendant vertex sits offset from + # its boundary vertex in the direction of its parent-graph + # incident edge (so the pendant visually replaces that edge). + incident_other = {v: [] for v in G_i.vertices()} + for e, ed in edge_depth.items(): + if ed == d: + continue + u, w = e + if u in incident_other: + incident_other[u].append((w, ed)) + if w in incident_other: + incident_other[w].append((u, ed)) + + pendant_pos = {} + OFFSET = 0.20 + for (v, pid) in out_spokes + in_spokes: + # Direction: toward the parent-graph neighbor at depth d-1 or d+1 + target_depth = d - 1 if pid[0] == 'out' else d + 1 + target = None + for (w, ed) in incident_other.get(v, []): + if ed == target_depth: + target = w + break + if target is not None and target in pos: + tx, ty = pos[target]; vx, vy = pos[v] + dx, dy = tx - vx, ty - vy + norm = (dx*dx + dy*dy) ** 0.5 or 1.0 + pendant_pos[pid] = (vx + OFFSET * dx / norm, + vy + OFFSET * dy / norm) + else: + vx, vy = pos[v] + pendant_pos[pid] = (vx + OFFSET, vy) + + # Draw spokes + for (v, pid) in out_spokes: + x1, y1 = pos[v]; x2, y2 = pendant_pos[pid] ax.plot([x1, x2], [y1, y2], color='#d62728', linewidth=2.4, linestyle='--', zorder=2) - for e in outer_set: - u, v = e - x1, y1 = pos[u]; x2, y2 = pos[v] + for (v, pid) in in_spokes: + x1, y1 = pos[v]; x2, y2 = pendant_pos[pid] ax.plot([x1, x2], [y1, y2], color='#2ca02c', linewidth=2.4, linestyle='--', zorder=2) @@ -130,14 +188,25 @@ def draw_cut_tire(ax, G_i, pos, edge_depth, tire, d, title): markeredgewidth=1.0) else: ax.plot(x, y, 'o', color='#bbbbbb', markersize=5, zorder=2) + # Draw pendant vertices + for (v, pid) in out_spokes: + px, py = pendant_pos[pid] + ax.plot(px, py, 's', color='#ffaa66', markersize=8, zorder=4, + markeredgecolor='#aa5500') + for (v, pid) in in_spokes: + px, py = pendant_pos[pid] + ax.plot(px, py, 's', color='#a0e0a0', markersize=8, zorder=4, + markeredgecolor='#2a7a2a') legend = [ plt.Line2D([], [], color='#1f77b4', linewidth=3, label=f'face boundary (depth {d})'), plt.Line2D([], [], color='#d62728', linewidth=2.4, - linestyle='--', label=f'inner spokes (depth {d - 1})'), + linestyle='--', + label=f'out spokes (toward depth {d - 1})'), plt.Line2D([], [], color='#2ca02c', linewidth=2.4, - linestyle='--', label=f'outer spokes (depth {d + 1})'), + linestyle='--', + label=f'in spokes (toward depth {d + 1})'), ] ax.legend(handles=legend, loc='upper left', bbox_to_anchor=(1.02, 1.0), fontsize=9, frameon=False) @@ -164,37 +233,13 @@ def main(): # Try several depths in G'_1 (which has depths 0-7) and pick a # decent example. print(f"G'_1 depths range: 0 to {max(ed1.values())}") - chosen_d = None - chosen_tire = None for d in range(1, max(ed1.values())): tires = cut_tire_at(H1, ed1, d) - print(f' depth {d}: {len(tires)} faces in depth-{d} subgraph') - for tire in tires: - print(f' face length {tire["face_length"]}, ' - f'inner spokes: {len(tire["inner_edges"])}, ' - f'outer spokes: {len(tire["outer_edges"])}') - # Pick the largest "interesting" face: try to find one with - # > 3 face edges AND inner/outer spokes both > 0. - candidates = [t for t in tires - if t['face_length'] >= 4 - and t['inner_edges'] and t['outer_edges']] - if candidates and chosen_tire is None: - chosen_d = d - # Pick the face with the most spokes total (most "informative") - candidates.sort(key=lambda t: -(len(t['inner_edges']) + - len(t['outer_edges']))) - chosen_tire = candidates[0] - - if chosen_tire is None: - print('No suitable cut tire found.') - return - - print(f"\nChosen cut tire at depth d = {chosen_d}") - print(f" face length: {chosen_tire['face_length']}") - print(f" inner spokes (depth {chosen_d - 1}): " - f"{len(chosen_tire['inner_edges'])}") - print(f" outer spokes (depth {chosen_d + 1}): " - f"{len(chosen_tire['outer_edges'])}") + print(f' depth {d}: {len(tires)} faces in H_d') + for t in tires: + print(f' face length {t["face_length"]}, ' + f'out spokes: {len(t["out_spokes"])}, ' + f'in spokes: {len(t["in_spokes"])}') # Collect "best" cut tires at multiple depths selected = [] @@ -202,10 +247,10 @@ def main(): tires = cut_tire_at(H1, ed1, d) candidates = [t for t in tires if t['face_length'] >= 4 - and t['inner_edges'] and t['outer_edges']] + and (t['out_spokes'] or t['in_spokes'])] if candidates: - candidates.sort(key=lambda t: -(len(t['inner_edges']) + - len(t['outer_edges']))) + candidates.sort(key=lambda t: -(len(t['out_spokes']) + + len(t['in_spokes']))) selected.append((d, candidates[0])) print(f"\nCut tires selected for visualization at depths: " f"{[d for d, _ in selected]}") @@ -225,18 +270,20 @@ def main(): draw_cut_tire(ax, H1, pos1, ed1, tire, d, title=(f"$d = {d}$: face length " f"{tire['face_length']}, " - f"{len(tire['inner_edges'])} inner + " - f"{len(tire['outer_edges'])} outer spokes")) + f"{len(tire['out_spokes'])} out + " + f"{len(tire['in_spokes'])} in spokes")) ax.set_xlim(min(xs) - 0.5, max(xs) + 0.5) ax.set_ylim(min(ys) - 0.5, max(ys) + 0.5) for idx in range(n, len(axes_flat)): axes_flat[idx].axis('off') fig.suptitle(r"Cut tires on $G'_1$ (Holton-McKay #0, V\S half) at " r"several depths $d$" + "\n" - r"Each panel: blue face boundary at depth $d$ " - r"+ inner spokes (depth $d-1$, red dashed) " - r"+ outer spokes (depth $d+1$, green dashed)", - fontsize=11, y=1.00) + r"Blue = face boundary at depth $d$ (in $H_d$). " + r"Orange-square pendants = out spokes (toward depth $d-1$). " + r"Green-square pendants = in spokes (toward depth $d+1$)." + "\n" + r"Each spoke is a labelled pendant added at a " + r"degree-$2$ boundary vertex.", + fontsize=10, y=1.00) plt.tight_layout() plt.savefig(out_fig, dpi=160, bbox_inches='tight') plt.close() diff --git a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux index 3179681..4bed7a9 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux +++ b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux @@ -4,9 +4,9 @@ \@writefile{toc}{\contentsline {paragraph}{Visualization.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Relation to the existing tire framework.}{2}{}\protected@file@percent } \@writefile{toc}{\contentsline {paragraph}{Example on $G'_1$ (Holton-McKay \#0, $V \setminus S$ half).}{3}{}\protected@file@percent } -\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{4}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Cut-and-depth-label procedure on Holton-McKay graph \#0 ($38$ vertices, $57$ edges, cubic planar non-Hamiltonian). The $6$-edge matching cut splits $G'$ into a $10$-vertex piece $G'_0$ (left) and a $28$-vertex piece $G'_1$ (right). Pendant edges (dashed, dark purple) are at depth $0$; the colour gradient encodes depth $0$ through max depth via BFS in the line-graph sense. Orange squares are the new pendant vertices; red circles are the boundary vertices in $V_i$; gray circles are interior vertices.\relax }}{5}{}\protected@file@percent } \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}} -\newlabel{fig:cut-depth-label}{{1}{4}} -\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{5}{}\protected@file@percent } -\newlabel{fig:cut-tire}{{2}{5}} -\gdef \@abspage@last{5} +\newlabel{fig:cut-depth-label}{{1}{5}} +\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Cut tires on $G'_1$ at depths $d = 1, 2, 4, 5, 6$. In each panel, blue solid edges form the face boundary at depth $d$; red dashed edges are inner spokes (depth $d - 1$); green dashed edges are outer spokes (depth $d + 1$). Vertices on the face boundary are highlighted; the rest of $G'_1$ is faded.\relax }}{6}{}\protected@file@percent } +\newlabel{fig:cut-tire}{{2}{6}} +\gdef \@abspage@last{6} diff --git a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.fdb_latexmk b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.fdb_latexmk new file mode 100644 index 0000000..58a5f76 --- /dev/null +++ b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.fdb_latexmk @@ -0,0 +1,84 @@ +# Fdb version 3 +["pdflatex"] 1779825367 "/Users/didericis/Code/math-research/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex" "cut_depth_label.pdf" "cut_depth_label" 1779825368 + "/Users/didericis/Code/math-research/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex" 1779825366 10920 bbdfc135e2020fd4e775aa257f60a3f1 "" + "/usr/local/texlive/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc" 1136849721 2900 1537cc8184ad1792082cd229ecc269f4 "" + 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zu*lrW^n=z!w@>NveP=s6B}7XuBEf{fLQ*NDSmc#Wqhu4{ut|%_WSkp3xW9Qpr9@lk zx@)V{;qQq8HSo)&@?=r+mxKplKeQzPaAYsEcPlJ^ZHS)lqQCh-OJ8 zv?Qpbkl2;vx+ch@Q0Uzq8hA5;0QhUsLjlbd%kX^6T7Dp-#g(sXGDGu_H)i!{t5)^c4vgsFqLtF?p~MJs@GeIKq#aZ78B{p1`{8ZJ7i zy61Fr%M;?Al8=o<9X|YX>N)MOTTy9lIF7+3j6Wvan_xOg4;Y0cfMw<2Vuv9o7gG?2 F`5%%qU0whH diff --git a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex index 79cf4d2..250196d 100644 --- a/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex +++ b/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex @@ -136,34 +136,58 @@ depth-$d$ edges). Equip $H_d$ with the planar embedding inherited from $G'_i$. For each face $f$ of $H_d$, the \emph{cut tire at $(d, f)$} is the -subgraph of $G'_i$ consisting of: +graph built as follows: \begin{itemize} - \item every edge on the boundary walk of $f$ (all of depth $d$ in - $H_d$), and - \item every edge of $G'_i$ with at least one endpoint on the - boundary walk of $f$ whose depth is $d - 1$ or $d + 1$. + \item Start with the boundary walk of $f$ as the \emph{face + boundary} (every edge here is of depth $d$ in $H_d$). + \item For each vertex $v$ on the boundary walk of $f$ that has + degree exactly $2$ in $H_d$: + \begin{itemize} + \item If $v$ is incident in $G'_i$ to an edge of depth + $d - 1$, add a fresh vertex $n_v$ and a fresh edge + $\{v, n_v\}$, labelled an \emph{out spoke}. + \item If $v$ is incident in $G'_i$ to an edge of depth + $d + 1$, add a fresh vertex $n_v$ and a fresh edge + $\{v, n_v\}$, labelled an \emph{in spoke}. + \end{itemize} \end{itemize} -The first set forms the \emph{face boundary at depth $d$}; the second -splits into \emph{inner spokes} (depth $d - 1$, pointing toward the -cut) and \emph{outer spokes} (depth $d + 1$, pointing away from the -cut). + +\noindent\emph{Convention.} ``Out'' means toward lower depth (toward +the cut, the outer boundary of $G'_i$); ``in'' means toward higher +depth (deeper into $G'_i$). Because $G'_i$ is cubic at every +original vertex, a degree-$2$ boundary vertex $v$ has exactly one +non-$H_d$ edge in $G'_i$, so $v$ contributes exactly one spoke (either +in or out, not both). + +\noindent\emph{Structural remark.} Under this definition, each cut +tire is intrinsically a graph of the form ``cycle (or closed walk) $+$ +pendants attached at simple boundary vertices,'' structurally +isomorphic to the partial tire dual $D(T)$ of \texttt{paper.tex} +(Proposition 1.8: $D(T) \cong C_{n+m} \circ K_1$ in the spoke-only +case). The labelled in/out spokes are the analogue of $D(T)$'s +leaves; the face boundary plays the role of $T'_{\mathrm{ann}}$. \end{definition} \paragraph{Relation to the existing tire framework.} Under the -correspondence between primal level structure (paper.tex) and dual -depth labelling, a cut tire on $G'_i$ at $(d, f)$ corresponds to: +correspondence between primal level structure (\texttt{paper.tex}) +and dual depth labelling, a cut tire on $G'_i$ at $(d, f)$ is +isomorphic as a graph to the partial tire dual $D(T)$ of a tire +$T$ in \texttt{paper.tex} (Def.~1.7): \begin{itemize} - \item face boundary at depth $d$ $\longleftrightarrow$ the tire - annular subgraph $T'_{\mathrm{ann}}$ at depth $d$ from the - cut (cf.\ Def.~1.15 of paper.tex); - \item the cut tire itself $\longleftrightarrow$ the tire annular - face connector $T'_{f'}$ (cf.\ Def.~1.16); - \item inner / outer spokes $\longleftrightarrow$ the inner and - outer spokes of $T'_{f'}$ (cf.\ Def.~1.17). + \item face boundary at depth $d$ $\longleftrightarrow$ + $T'_{\mathrm{ann}}$, the cycle (or closed walk) of annular + dual vertices; + \item in/out spokes $\longleftrightarrow$ the leaves of $D(T)$ + (Def.~1.7's degree-$1$ pendants attached to the cycle); + \item the cut tire's planar embedding $\longleftrightarrow$ + $D(T)$'s embedding. \end{itemize} -So the cut tire is the dual-side analogue of the -``tire annular face connector,'' parametrised by depth from the cut -rather than depth from a primal level source. +The cut tire is therefore not just analogous to but \emph{is} a +partial tire dual (up to relabelling), with depth-from-cut playing +the role of level depth from the primal source. Consequently every +proposition about $D(T)$ in paper.tex --- chromatic polynomial +counts, $S_3$-orbit structure, rainbow conjecture, etc.\ --- applies +verbatim to each cut tire. \paragraph{Example on $G'_1$ (Holton-McKay \#0, $V \setminus S$ half).} @@ -178,17 +202,20 @@ highlighted; the rest of $G'_1$ is faded.} \label{fig:cut-tire} \end{figure} -In this example: +In this example, counting only degree-$2$ boundary vertices (which +under the redefinition each contribute exactly one in/out spoke): \begin{itemize} - \item $d = 1$: face length $12$, $5$ inner spokes (to depth-$0$ - pendants) $+$ $4$ outer spokes. This is the outermost cut - tire, immediately adjacent to the cut. + \item $d = 1$: face length $12$, $5$ out spokes (toward depth-$0$ + pendants) $+\ 0$ in spokes. The outermost cut tire, + immediately adjacent to the cut. No in spokes because every + depth-$2$ neighbour of a face-boundary vertex is at a + higher-degree vertex of $H_1$. \item $d = 2$: face length $7$ (one of two symmetric faces in - $H_2$), $4 + 3$ spokes. - \item $d = 4$: face length $8$, $2 + 5$ spokes. - \item $d = 5$: face length $14$, $4 + 6$ spokes. - \item $d = 6$: face length $12$, $7 + 1$ spokes. This is the - innermost cut tire (one face left, almost no outer spokes). + $H_2$), $4$ out $+\ 3$ in spokes. + \item $d = 4$: face length $8$, $2$ out $+\ 1$ in. + \item $d = 5$: face length $14$, $4$ out $+\ 2$ in. + \item $d = 6$: face length $12$, $3$ out $+\ 2$ in. Innermost cut + tire in this chain. \end{itemize} \section*{Connection to chain pigeonhole / 4CT reducibility} diff --git a/papers/coloring_nested_tire_graphs/notes/fig_cut_tire.png b/papers/coloring_nested_tire_graphs/notes/fig_cut_tire.png index b4a2bd2afec24f12987dfae49a39db84b4763e16..e8016ec13a5a5b43ef383c2e5c9c83eeb19344f5 100644 GIT binary patch literal 232836 zcmdpeWmr^e`!5D4Di~ORU?CmS(juXPgpwi*DBU^4z<`M&5{iH{f|SHi(l9D0odXO# zD4ioQbewy+_xqmzIiJs`!*%Tq!;HhMwVvm`e|4__P4$~}hnNmgQBl#|zNL7Nit3;m z6&0=3K^pi?jBsx`wD(X7fXTrn^scrF0@X!?z>V^osB{M+jATJ>^;>UDyrLx^4gxU zvx5g>8Gh{UE___2?@%mT9+J?%rTJAQ?@==6 z>&FMhj!j*toZVfBC;Kfm4qhoWBh2&TCPt2bpR@3=wB(2xjB0Hebvain2!H&)UtH`S z`3(Pk1V4lx*$)T&->=8-72Edz_YoDfJ$=lw>$8qQ!t290LKsDD-q&+;23~(OHGtF8UGv2mmUhIVyYHvEO?9w)%nz&- zxK6Z)rl^DqrNz3a+~{}}>pIlRkLL*3qF0`&VbCbF-d<{z3yV7mC-cJCF%h`dk9nuy zSK_`p#HGxc#%%f>v4EmAU>iVRLulrF-whUTdH2`@+N3SY0l+ zvsOKTnaNHyDtAw{>&<63uJS2bz$pcuwrKhwI8M-zqbr+*_jL?DC-EpUAd=ak)KA99KWaPWEwLV@*P%ofcRC;|#&o7y 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