From 5fa6e9e8403eebd0c942af7501fcb791686a4a68 Mon Sep 17 00:00:00 2001 From: didericis Date: Mon, 25 May 2026 20:52:50 -0400 Subject: [PATCH] coloring_nested_tire_graphs: switch bridge example to barbell (non-pendant bridge) The previous bridge example used a pendant edge as the bridge. A pendant edge IS technically a bridge (single-edge cut), but the intended notion was a "proper" non-pendant bridge: an edge cut connecting two non-trivial subgraphs. Replaced with the smallest example: - B_out = 4-cycle on {0, 1, 2, 3}. - O = barbell on {4..9}: two disjoint triangles {4, 5, 6} and {7, 8, 9} connected by the bridge edge 6-7. - Annular triangulation by hand (12 triangles, all listed in the generator script with planarity verified by Sage). The barbell case is structurally cleaner: BOTH endpoints of the bridge have degree >= 2 in O, and the interior dual subgraph has the two bridge-incident annular faces (d_5, d_6) as its trivalent theta-graph vertices (in the pendant case, the trivalent vertices were NOT bridge-incident, which was confusing). Updates fig_partial_tire_dual_bridge.png and the figure caption. Paper stays at 9 pages. Co-Authored-By: Claude Opus 4.7 --- .../draw_partial_tire_dual_bridge.py | 258 ++++++++++-------- .../experiments/partial_tire_dual_bridge.png | Bin 154565 -> 146308 bytes .../fig_partial_tire_dual_bridge.png | Bin 154565 -> 146308 bytes papers/coloring_nested_tire_graphs/paper.aux | 6 +- papers/coloring_nested_tire_graphs/paper.log | 50 ++-- papers/coloring_nested_tire_graphs/paper.pdf | Bin 748857 -> 746924 bytes papers/coloring_nested_tire_graphs/paper.tex | 19 +- 7 files changed, 178 insertions(+), 155 deletions(-) diff --git a/papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual_bridge.py b/papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual_bridge.py index eae5039..7d5abdd 100644 --- a/papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual_bridge.py +++ b/papers/coloring_nested_tire_graphs/experiments/draw_partial_tire_dual_bridge.py @@ -1,17 +1,18 @@ """Draw a partial tire dual D(T) for a tire whose inner outerplanar -graph O has a bridge. +graph O has a (non-pendant) bridge: a single edge whose removal +disconnects O into two non-trivial components. Tire construction: - - Outer cycle B_out: triangle on {0, 1, 2}. - - Inner outerplanar O on {3, 4, 5, 6}: triangle 3-4-5 plus pendant - edge 5-6 (the bridge of O). - - Annular triangulation with 8 triangles (computed by hand below). + - Outer cycle B_out: 4-cycle on {0, 1, 2, 3}. + - Inner outerplanar O on {4..9}: two disjoint triangles {4, 5, 6} + and {7, 8, 9}, connected by the bridge edge 6-7. + - Annular triangulation built by hand below. -The bridge 5-6 has both incident faces in the annular region, so in -the partial tire dual it contributes an interior dual edge (not a -leaf). This makes the interior dual subgraph a theta graph rather -than a single cycle: two trivalent vertices (the two annular faces -incident to the bridge) connected by three paths. +The bridge 6-7 has both incident faces in the annular region, so in +the partial tire dual it contributes an interior dual edge (not +leaves). This makes the interior dual subgraph a theta graph: two +trivalent vertices (the two annular faces incident to the bridge, +or those with all interior edges) connected by three paths. """ import math import os @@ -21,54 +22,72 @@ from collections import defaultdict import matplotlib.pyplot as plt import matplotlib.patches as patches +from sage.all import Graph + # Explicit vertex positions POS = { - 0: (0.0, 1.0), - 1: (-0.866, -0.5), - 2: (0.866, -0.5), - 3: (0.0, 0.32), - 4: (-0.27, -0.17), - 5: (0.13, -0.17), - 6: (0.52, -0.07), + 0: (0.0, 1.6), # outer top + 1: (-1.6, 0.0), # outer left + 2: (0.0, -1.6), # outer bottom + 3: (1.6, 0.0), # outer right + 4: (-0.65, 0.42), # left triangle, top + 5: (-0.65, -0.42), # left triangle, bottom + 6: (-0.20, 0.0), # left triangle, right (= bridge endpoint 1) + 7: (0.20, 0.0), # right triangle, left (= bridge endpoint 2) + 8: (0.65, 0.42), # right triangle, top + 9: (0.65, -0.42), # right triangle, bottom } -# Annular triangles (in lattice-path-like order, but here just listed) -TRIANGLES = [ - (0, 1, 4), # T1 -- O-move - (0, 4, 3), # T2 -- has edge 4-3 boundary (inner triangle) - (1, 2, 5), # T3 -- O-move - (1, 5, 4), # T4 -- inner-triangle-edge boundary - (2, 0, 6), # T5 -- O-move - (0, 3, 6), # T6 -- ALL INTERIOR (bridge neighbour) - (3, 5, 6), # T7 -- inner-triangle-edge boundary, contains bridge - (5, 2, 6), # T8 -- ALL INTERIOR (bridge neighbour) -] - -# Tire edges (all) T_EDGES = [ - # outer cycle (B_out) - (0, 1), (1, 2), (2, 0), - # O: triangle 3-4-5 - (3, 4), (4, 5), (3, 5), - # O: pendant / bridge - (5, 6), - # spokes - (0, 3), (1, 4), (2, 5), - # diagonals from annular triangulation - (0, 4), (1, 5), (0, 6), (2, 6), (3, 6), + # outer cycle B_out (length 4) + (0, 1), (1, 2), (2, 3), (3, 0), + # O: triangle 4-5-6 + (4, 5), (5, 6), (4, 6), + # O: triangle 7-8-9 + (7, 8), (8, 9), (7, 9), + # O: bridge + (6, 7), + # annular edges (spokes + diagonals chosen to triangulate) + (0, 4), (0, 6), (0, 8), + (1, 4), (1, 5), + (2, 5), (2, 6), (2, 7), (2, 9), + (3, 8), (3, 9), + (0, 7), # extra diagonal to triangulate top ] -# Boundary edges -B_OUT_EDGES = [(0, 1), (1, 2), (2, 0)] -# Edges of O on inner-triangle-face boundary (i.e., NOT the bridge) -B_IN_EDGES = [(3, 4), (4, 5), (3, 5)] -# Bridge edge (interior to annulus) -BRIDGE = (5, 6) +# Annular triangles -- listed explicitly so we know exactly which +# faces are which. 12 of them. +TRIANGLES = [ + (0, 1, 4), # top-left top + (1, 4, 5), # left side, with triangle 1 + (1, 2, 5), # bottom-left + (2, 5, 6), # right of left-triangle, bottom + (0, 4, 6), # right of left-triangle, top + (0, 6, 7), # top-center over the bridge (bridge-incident) + (2, 6, 7), # bottom-center over the bridge (bridge-incident) + (0, 7, 8), # right of right-triangle, top + (2, 7, 9), # right of right-triangle, bottom + (0, 8, 3), # top-right + (3, 8, 9), # right side, with triangle 2 + (2, 9, 3), # bottom-right +] + +# Boundary edges: +B_OUT_EDGES = [(0, 1), (1, 2), (2, 3), (3, 0)] +# Edges of O on inner-triangle-face boundaries (= NOT the bridge): +B_IN_EDGES = [(4, 5), (5, 6), (4, 6), + (7, 8), (8, 9), (7, 9)] +BRIDGE = (6, 7) def main(): - # Map: edge -> list of triangle indices containing it + # Sanity-check via sage that the graph is planar + G = Graph(T_EDGES) + assert G.is_planar(), "Constructed tire is not planar" + print(f"Tire graph: n={G.order()}, m={G.size()}, planar=True") + + # Map: edge -> triangle indices containing it edge_to_tris = defaultdict(list) for i, t in enumerate(TRIANGLES): for e in (frozenset({t[0], t[1]}), @@ -76,67 +95,84 @@ def main(): frozenset({t[0], t[2]})): edge_to_tris[e].append(i) - # Verify each B_out edge is in exactly 1 annular triangle + # Verify each B_out edge in exactly 1 annular triangle for u, v in B_OUT_EDGES: ts = edge_to_tris[frozenset({u, v})] - assert len(ts) == 1, f"B_out edge {u}-{v} in {ts}" - - # Each B_in (= non-bridge O edge) is in exactly 1 annular triangle + assert len(ts) == 1, f"B_out edge {u}-{v} in {ts} (expected 1)" + # Each B_in (non-bridge O edge) in exactly 1 annular triangle for u, v in B_IN_EDGES: ts = edge_to_tris[frozenset({u, v})] - assert len(ts) == 1, f"B_in edge {u}-{v} in {ts}" - - # Bridge is in exactly 2 annular triangles (interior to annulus) + assert len(ts) == 1, f"B_in edge {u}-{v} in {ts} (expected 1)" + # Bridge in exactly 2 annular triangles bts = edge_to_tris[frozenset(BRIDGE)] print(f"Bridge {BRIDGE} is in annular triangles {bts}") assert len(bts) == 2 - # Interior dual subgraph: connect annular triangles sharing an edge + # Verify every edge of T is accounted for + seen_in_tris = set() + for t in TRIANGLES: + for e in (frozenset({t[0], t[1]}), + frozenset({t[1], t[2]}), + frozenset({t[0], t[2]})): + seen_in_tris.add(e) + all_T_edges = {frozenset(e) for e in T_EDGES} + # Each edge appears in at least one triangle EXCEPT it might be that + # boundary edges to non-annular faces are accounted for differently. + # Annular triangles use boundary edges (B_out and inner-triangle-of-O + # edges) plus interior annular edges. Every edge of T should appear + # in at least one annular triangle EXCEPT the inner faces of O don't + # exist as separate annular triangles. Wait, the inner triangle + # faces of O (= {4,5,6} and {7,8,9}) are themselves face boundaries + # of T but NOT in the annular region. So edges of the inner + # triangles of O appear in exactly ONE annular triangle (across the + # boundary) + the inner triangle face on the other side. + # Check: every T edge should appear in at most 2 annular triangles + # (and at least 1 if it's incident to any annular face). + for e in all_T_edges: + n = len(edge_to_tris[e]) + assert n in (1, 2), \ + f"Edge {sorted(e)} appears in {n} annular triangles" + + # Interior dual adjacencies n_tri = len(TRIANGLES) tri_adj = defaultdict(set) for e, ts in edge_to_tris.items(): if len(ts) == 2: tri_adj[ts[0]].add(ts[1]) tri_adj[ts[1]].add(ts[0]) - print("Interior dual subgraph degree sequence:") for i in range(n_tri): - print(f" T{i+1} ({TRIANGLES[i]}): degree {len(tri_adj[i])}") + print(f" T{i} {TRIANGLES[i]}: degree {len(tri_adj[i])}") - # Centroids of triangles for d_f positions - centroids = [] - for t in TRIANGLES: - cx = (POS[t[0]][0] + POS[t[1]][0] + POS[t[2]][0]) / 3 - cy = (POS[t[0]][1] + POS[t[1]][1] + POS[t[2]][1]) / 3 - centroids.append((cx, cy)) + # Centroids of triangles (for d_f positions) + centroids = [((POS[t[0]][0] + POS[t[1]][0] + POS[t[2]][0]) / 3, + (POS[t[0]][1] + POS[t[1]][1] + POS[t[2]][1]) / 3) + for t in TRIANGLES] # === Plot === fig, ax = plt.subplots(figsize=(10, 9)) - # Draw underlying tire edges faintly - outer_set = {0, 1, 2} - inner_set = {3, 4, 5, 6} + outer_set = {0, 1, 2, 3} + inner_set = {4, 5, 6, 7, 8, 9} + bridge_fs = frozenset(BRIDGE) for u, v in T_EDGES: x1, y1 = POS[u]; x2, y2 = POS[v] - if u in outer_set and v in outer_set: - color = '#a8c9e8'; lw = 2.0 + if frozenset({u, v}) == bridge_fs: + color, lw = '#cc6677', 2.8 # bridge of O + elif u in outer_set and v in outer_set: + color, lw = '#a8c9e8', 2.0 elif u in inner_set and v in inner_set: - # Distinguish bridge (pendant) from triangle edges - if frozenset({u, v}) == frozenset(BRIDGE): - color = '#cc6677'; lw = 2.5 # darker red for bridge - else: - color = '#e8a8a8'; lw = 2.0 + color, lw = '#e8a8a8', 2.0 else: - color = '#cccccc'; lw = 1.0 + color, lw = '#cccccc', 1.0 ax.plot([x1, x2], [y1, y2], color=color, linewidth=lw, zorder=1) - # Tire vertices - for v in [0, 1, 2]: + for v in outer_set: x, y = POS[v] - ax.plot(x, y, 'o', color='#a8c9e8', markersize=15, zorder=2) + ax.plot(x, y, 'o', color='#a8c9e8', markersize=16, zorder=2) ax.annotate(str(v), (x, y), color='white', ha='center', va='center', fontsize=9, fontweight='bold', zorder=3) - for v in [3, 4, 5, 6]: + for v in inner_set: x, y = POS[v] ax.plot(x, y, 'o', color='#e8a8a8', markersize=13, zorder=2) ax.annotate(str(v), (x, y), color='white', ha='center', va='center', @@ -144,24 +180,22 @@ def main(): DUAL_COLOR = '#7d3c98' LEAF_COLOR = '#e67e22' - BRIDGE_DUAL_COLOR = '#cc4444' # color the bridge's dual edge differently + BRIDGE_DUAL_COLOR = '#cc4444' # Interior dual edges for i in range(n_tri): for j in tri_adj[i]: - if j <= i: - continue - x1, y1 = centroids[i]; x2, y2 = centroids[j] - # Find shared edge to colour-code bridge edge + if j <= i: continue ti_es = {frozenset({TRIANGLES[i][a], TRIANGLES[i][b]}) for a, b in [(0, 1), (1, 2), (0, 2)]} tj_es = {frozenset({TRIANGLES[j][a], TRIANGLES[j][b]}) for a, b in [(0, 1), (1, 2), (0, 2)]} shared = (ti_es & tj_es).pop() - if shared == frozenset(BRIDGE): + if shared == bridge_fs: ec, lw = BRIDGE_DUAL_COLOR, 3.0 else: ec, lw = DUAL_COLOR, 2.0 + x1, y1 = centroids[i]; x2, y2 = centroids[j] ax.plot([x1, x2], [y1, y2], color=ec, linewidth=lw, zorder=4) # Leaves for B_out edges @@ -170,18 +204,15 @@ def main(): midx = (POS[u][0] + POS[v][0]) / 2 midy = (POS[u][1] + POS[v][1]) / 2 d = math.sqrt(midx**2 + midy**2) - push = 1.20 + push = 1.18 lpos = (midx * push / d, midy * push / d) cx, cy = centroids[ti] ax.plot([cx, lpos[0]], [cy, lpos[1]], color=LEAF_COLOR, linewidth=1.5, linestyle='--', zorder=4) - ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=11, + ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=10, zorder=5) - ax.annotate(f"$\\ell^{{\\mathrm{{out}}}}_{{{u},{v}}}$", lpos, - color='white', ha='center', va='center', fontsize=6, - fontweight='bold', zorder=6) - # Leaves for non-bridge B_in edges (= the 3 inner triangle edges) + # Leaves for non-bridge edges of O (= inner triangle edges) for u, v in B_IN_EDGES: ti = edge_to_tris[frozenset({u, v})][0] midx = (POS[u][0] + POS[v][0]) / 2 @@ -190,30 +221,21 @@ def main(): vx, vy = midx - cx, midy - cy norm = math.sqrt(vx*vx + vy*vy) offset = 0.10 - if norm > 1e-9: - nx, ny = vx / norm, vy / norm - else: - nx, ny = 0, -1 + nx, ny = (vx / norm, vy / norm) if norm > 1e-9 else (0, -1) lpos = (midx + nx * offset, midy + ny * offset) ax.plot([cx, lpos[0]], [cy, lpos[1]], color=LEAF_COLOR, linewidth=1.5, linestyle='--', zorder=4) - ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=11, + ax.plot(lpos[0], lpos[1], 'D', color=LEAF_COLOR, markersize=10, zorder=5) - ax.annotate(f"$\\ell^{{\\mathrm{{in}}}}_{{{u},{v}}}$", lpos, - color='white', ha='center', va='center', fontsize=6, - fontweight='bold', zorder=6) - # Interior dual vertices d_f, highlighting the two degree-3 - # vertices in a darker color (these are the trivalent vertices - # of the theta graph that the interior dual forms). + # Interior dual vertices, highlight degree-3 (trivalent) for ti, (cx, cy) in enumerate(centroids): - is_deg3 = (len(tri_adj[ti]) == 3) - marker_color = '#5a2273' if is_deg3 else DUAL_COLOR - ax.plot(cx, cy, 's', color=marker_color, markersize=15, zorder=5) + deg = len(tri_adj[ti]) + c = '#5a2273' if deg == 3 else DUAL_COLOR + ax.plot(cx, cy, 's', color=c, markersize=14, zorder=5) ax.annotate(f"$d_{{{ti}}}$", (cx, cy), color='white', ha='center', - va='center', fontsize=7, fontweight='bold', zorder=6) + va='center', fontsize=6, fontweight='bold', zorder=6) - # Legend legend_items = [ plt.Line2D([], [], marker='o', color='w', markerfacecolor='#a8c9e8', markersize=12, label='$B_{\\mathrm{out}}$ vertex'), @@ -223,13 +245,13 @@ def main(): label='$B_{\\mathrm{out}}$ edge'), plt.Line2D([], [], color='#e8a8a8', linewidth=2.0, label='triangle edge of $O$'), - plt.Line2D([], [], color='#cc6677', linewidth=2.5, - label='bridge of $O$ (= pendant)'), + plt.Line2D([], [], color='#cc6677', linewidth=2.8, + label='bridge of $O$'), plt.Line2D([], [], marker='s', color='w', markerfacecolor=DUAL_COLOR, - markersize=12, label='$d_f$ (degree 2 in interior dual)'), + markersize=12, label='$d_f$ (deg 2 in interior dual)'), plt.Line2D([], [], marker='s', color='w', markerfacecolor='#5a2273', markersize=12, - label='$d_f$ (degree 3 -- theta-graph trivalent)'), + label='$d_f$ (deg 3 -- trivalent theta-graph vertex)'), plt.Line2D([], [], marker='D', color='w', markerfacecolor=LEAF_COLOR, markersize=10, label='leaf'), plt.Line2D([], [], color=DUAL_COLOR, linewidth=2.0, @@ -242,16 +264,18 @@ def main(): ax.legend(handles=legend_items, loc='upper left', bbox_to_anchor=(1.02, 1.0), fontsize=8, frameon=False) - ax.set_xlim(-1.30, 1.50) - ax.set_ylim(-1.20, 1.20) + ax.set_xlim(-2.10, 2.60) + ax.set_ylim(-1.90, 1.90) ax.set_aspect('equal') ax.axis('off') ax.set_title( - 'Partial tire dual $D(T)$ when $O$ has a bridge\n' - '$O$ = triangle $\\{3,4,5\\}$ + pendant edge $5$--$6$;\n' - 'interior dual is a theta graph (2 deg-3 vertices + 3 paths), ' - 'not a cycle.', - fontsize=11) + '$D(T)$ when $O$ is a barbell (two triangles + one bridge edge)\n' + '$O$ = triangle $\\{4,5,6\\}$ $\\cup$ bridge $6$--$7$ $\\cup$ ' + 'triangle $\\{7,8,9\\}$;\n' + 'bridge is interior to the annulus, so it contributes an interior ' + 'dual edge (red).\n' + 'Interior dual is a theta graph (2 trivalent $d_f$ + 3 paths).', + fontsize=10) HERE = os.path.dirname(os.path.abspath(__file__)) out = os.path.join(HERE, 'partial_tire_dual_bridge.png') diff --git a/papers/coloring_nested_tire_graphs/experiments/partial_tire_dual_bridge.png b/papers/coloring_nested_tire_graphs/experiments/partial_tire_dual_bridge.png index 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The ten interior vertices $d_f$ at the centroids of the annular triangles form a single $10$-cycle (solid purple); each boundary edge of the annular region (either of $B_{\mathrm {out}}$ or of $B_{\mathrm {in}}$) contributes a degree-$1$ leaf (orange diamond) attached to the unique annular face incident to it (dashed orange), giving the structure $C_{10} \circ K_1$ of Proposition\nonbreakingspace 1.8\hbox {}.}}{4}{}\protected@file@percent } \newlabel{fig:partial-tire-dual-example}{{3}{4}} \newlabel{prop:partial-tire-dual-structure}{{1.8}{4}} -\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge. Here $B_{\mathrm {out}}$ is a triangle on $\{0,1,2\}$ and $O$ is a triangle $\{3,4,5\}$ with a pendant edge $5$--$6$ (the bridge of $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.8\hbox {}, but a theta graph (two trivalent vertices $d_5, d_7$ connected by three internally vertex-disjoint paths in $D(T)$). Leaves come only from $B_{\mathrm {out}}$ ($n = 3$ leaves) and from the three non-bridge edges of $O$ (the three triangle edges of the inner triangle).}}{5}{}\protected@file@percent } +\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces Partial tire dual $D(T)$ when the inner outerplanar graph $O$ has a bridge --- here a non-trivial edge cut connecting two disjoint triangles. $B_{\mathrm {out}}$ is a $4$-cycle on $\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph {interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition\nonbreakingspace 1.8\hbox {}, but a theta graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident annular faces) are joined by three internally vertex-disjoint paths in $D(T)$. Leaves come only from $B_{\mathrm {out}}$ ($n = 4$ leaves) and the six non-bridge edges of $O$ ($m_{\partial } = 6$ leaves, three for each triangle).}}{5}{}\protected@file@percent } \newlabel{fig:partial-tire-dual-bridge}{{4}{5}} \newlabel{prop:no-level-d-pinch}{{1.9}{5}} \citation{bauerfeld-pds} \newlabel{lem:tire-component}{{1.10}{6}} \citation{bauerfeld-pds} -\newlabel{rem:tire-component-degenerate}{{1.11}{7}} +\newlabel{rem:tire-component-degenerate}{{1.11}{8}} \newlabel{rem:tire-no-extra-hypotheses}{{1.12}{8}} \newlabel{prop:edge-vertex-bijection}{{1.13}{8}} -\newlabel{rem:edge-vertex-corollary}{{1.14}{8}} \bibcite{bauerfeld-pds}{1} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{0pt} \newlabel{tocindent3}{0pt} +\newlabel{rem:edge-vertex-corollary}{{1.14}{9}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{9}{}\protected@file@percent } \gdef \@abspage@last{9} diff --git a/papers/coloring_nested_tire_graphs/paper.log b/papers/coloring_nested_tire_graphs/paper.log index 11c1f35..187d619 100644 --- a/papers/coloring_nested_tire_graphs/paper.log +++ b/papers/coloring_nested_tire_graphs/paper.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 20:43 +This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 25 MAY 2026 20:52 entering extended mode restricted \write18 enabled. %&-line parsing enabled. @@ -213,22 +213,18 @@ File: fig_partial_tire_dual.png Graphic file (type png) Package pdftex.def Info: fig_partial_tire_dual.png used on input line 225. 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a/papers/coloring_nested_tire_graphs/paper.tex +++ b/papers/coloring_nested_tire_graphs/paper.tex @@ -237,20 +237,23 @@ Proposition~\ref{prop:partial-tire-dual-structure}.} \begin{figure}[h] \centering -\includegraphics[width=0.78\textwidth]{fig_partial_tire_dual_bridge.png} +\includegraphics[width=0.85\textwidth]{fig_partial_tire_dual_bridge.png} \caption{Partial tire dual $D(T)$ when the inner outerplanar graph -$O$ has a bridge. Here $B_{\mathrm{out}}$ is a triangle on -$\{0,1,2\}$ and $O$ is a triangle $\{3,4,5\}$ with a pendant edge -$5$--$6$ (the bridge of $O$). Because both faces incident to the +$O$ has a bridge --- here a non-trivial edge cut connecting two +disjoint triangles. $B_{\mathrm{out}}$ is a $4$-cycle on +$\{0,1,2,3\}$ and $O$ is the barbell: triangle $\{4,5,6\}$ together +with triangle $\{7,8,9\}$ joined by the bridge edge $6$--$7$ (removing +the bridge disconnects $O$). Because both faces incident to the bridge are annular triangles, the bridge contributes an \emph{interior dual edge} (highlighted in red) rather than two leaves; consequently the interior dual subgraph is no longer the single $(n+m)$-cycle of Proposition~\ref{prop:partial-tire-dual-structure}, but a theta -graph (two trivalent vertices $d_5, d_7$ connected by three -internally vertex-disjoint paths in $D(T)$). Leaves come only from -$B_{\mathrm{out}}$ ($n = 3$ leaves) and from the three non-bridge -edges of $O$ (the three triangle edges of the inner triangle).} +graph: the two trivalent vertices $d_5, d_6$ (the bridge-incident +annular faces) are joined by three internally vertex-disjoint paths +in $D(T)$. Leaves come only from $B_{\mathrm{out}}$ ($n = 4$ leaves) +and the six non-bridge edges of $O$ ($m_{\partial} = 6$ leaves, +three for each triangle).} \label{fig:partial-tire-dual-bridge} \end{figure}