didericis/math-research
1
0
Fork 0
Code Issues Pull Requests Packages Projects Releases Wiki Activity

coloring_nested_tire_graphs: redefine cut tire bullet 2 as labelled pendants

Browse Source 
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 <noreply@anthropic.com>
This commit is contained in:
didericis
2026-05-26 15:56:26 -04:00
parent a21aff7f65
commit cb6a79f799
8 changed files with 637 additions and 134 deletions
Download Patch File Download Diff File Expand all files Collapse all files
papers/coloring_nested_tire_graphs/experiments/cut_tire.py
+128 -81
View File
@@ -40,13 +40,15 @@ def faces_of_subgraph(H, pos):
def cut_tire_at(G_i, edge_depth, d): def cut_tire_at(G_i, edge_depth, d):
"""Build cut tires at depth d for each face of the depth-d subgraph. """Build cut tires at depth d for each face of H_d.
Returns list of (face_vertices, tire_subgraph_edges_by_role) where:
tire_subgraph_edges_by_role is a dict with keys Per the (redefined) Definition, each cut tire is:
'face' (boundary at depth d), 'inner' (adjacent at d-1), - The face boundary walk of f (edges of depth d in H_d), AND
'outer' (adjacent at d+1). - 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_edges = [e for e, ed in edge_depth.items() if ed == d]
H_d = Graph([(u, v) for (u, v) in H_d_edges], H_d = Graph([(u, v) for (u, v) in H_d_edges],
multiedges=False, loops=False) multiedges=False, loops=False)
@@ -56,68 +58,124 @@ def cut_tire_at(G_i, edge_depth, d):
return [] return []
faces_h_d = H_d.faces() faces_h_d = H_d.faces()
results = [] results = []
for face in faces_h_d: # For each vertex in H_d, compute its degree in H_d and find its
# face is a list of (u, v) edge-pairs going around the boundary # incident non-H_d edges in G'_i (with their depths).
# in Sage; convert to vertex sequence h_d_degree = {v: H_d.degree(v) for v in H_d.vertices()}
vertices_on_face = [] incident_other = {v: [] for v in H_d.vertices()}
for (u, v) in face: for e, ed in edge_depth.items():
vertices_on_face.append(u) if ed == d:
face_vertex_set = set(vertices_on_face)
# Boundary edges of the face (in H_d): they're the face edges
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):
continue continue
u, v = e_full u, w = e
if u in face_vertex_set or v in face_vertex_set: 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:
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]
# 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
# 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: if ed == d - 1:
inner_edges.append(e_full) pendant_id = ('out', v)
else: pendant_to_boundary[pendant_id] = v
outer_edges.append(e_full) 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({ results.append({
'face_vertices': vertices_on_face, 'face_vertices_unique': face_vertex_list,
'face_edges': face_edges, 'face_edges': face_edges,
'inner_edges': inner_edges, 'out_spokes': out_spokes, # (boundary_v, pendant_id) for d-1
'outer_edges': outer_edges, 'in_spokes': in_spokes, # (boundary_v, pendant_id) for d+1
'face_length': len(face), 'face_length': len(face),
}) })
return results return results
def draw_cut_tire(ax, G_i, pos, edge_depth, tire, d, title): 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']) face_set = set(tire['face_edges'])
inner_set = set(tire['inner_edges']) face_vertex_set = set(tire['face_vertices_unique'])
outer_set = set(tire['outer_edges']) out_spokes = tire['out_spokes']
face_vertex_set = set(tire['face_vertices']) in_spokes = tire['in_spokes']
# Fade all G_i edges # Fade all G_i edges
for e in G_i.edges(labels=False): for e in G_i.edges(labels=False):
u, v = e u, v = e
e_norm = (min(u, v), max(u, v)) 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 continue
x1, y1 = pos[u]; x2, y2 = pos[v] x1, y1 = pos[u]; x2, y2 = pos[v]
ax.plot([x1, x2], [y1, y2], color='#dddddd', linewidth=0.9, ax.plot([x1, x2], [y1, y2], color='#dddddd', linewidth=0.9,
zorder=1) zorder=1)
# Highlight the tire # Highlight the face boundary
for e in face_set: for e in face_set:
u, v = e u, v = e
x1, y1 = pos[u]; x2, y2 = pos[v] x1, y1 = pos[u]; x2, y2 = pos[v]
ax.plot([x1, x2], [y1, y2], color='#1f77b4', linewidth=3.0, ax.plot([x1, x2], [y1, y2], color='#1f77b4', linewidth=3.0,
zorder=3, label=None) zorder=3)
for e in inner_set:
u, v = e # Compute pendant positions: each pendant vertex sits offset from
x1, y1 = pos[u]; x2, y2 = pos[v] # 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, ax.plot([x1, x2], [y1, y2], color='#d62728', linewidth=2.4,
linestyle='--', zorder=2) linestyle='--', zorder=2)
for e in outer_set: for (v, pid) in in_spokes:
u, v = e x1, y1 = pos[v]; x2, y2 = pendant_pos[pid]
x1, y1 = pos[u]; x2, y2 = pos[v]
ax.plot([x1, x2], [y1, y2], color='#2ca02c', linewidth=2.4, ax.plot([x1, x2], [y1, y2], color='#2ca02c', linewidth=2.4,
linestyle='--', zorder=2) linestyle='--', zorder=2)
@@ -130,14 +188,25 @@ def draw_cut_tire(ax, G_i, pos, edge_depth, tire, d, title):
markeredgewidth=1.0) markeredgewidth=1.0)
else: else:
ax.plot(x, y, 'o', color='#bbbbbb', markersize=5, zorder=2) 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 = [ legend = [
plt.Line2D([], [], color='#1f77b4', linewidth=3, plt.Line2D([], [], color='#1f77b4', linewidth=3,
label=f'face boundary (depth {d})'), label=f'face boundary (depth {d})'),
plt.Line2D([], [], color='#d62728', linewidth=2.4, 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, 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', ax.legend(handles=legend, loc='upper left',
bbox_to_anchor=(1.02, 1.0), fontsize=9, frameon=False) 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 # Try several depths in G'_1 (which has depths 0-7) and pick a
# decent example. # decent example.
print(f"G'_1 depths range: 0 to {max(ed1.values())}") 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())): for d in range(1, max(ed1.values())):
tires = cut_tire_at(H1, ed1, d) tires = cut_tire_at(H1, ed1, d)
print(f' depth {d}: {len(tires)} faces in depth-{d} subgraph') print(f' depth {d}: {len(tires)} faces in H_d')
for tire in tires: for t in tires:
print(f' face length {tire["face_length"]}, ' print(f' face length {t["face_length"]}, '
f'inner spokes: {len(tire["inner_edges"])}, ' f'out spokes: {len(t["out_spokes"])}, '
f'outer spokes: {len(tire["outer_edges"])}') f'in spokes: {len(t["in_spokes"])}')
# 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'])}")
# Collect "best" cut tires at multiple depths # Collect "best" cut tires at multiple depths
selected = [] selected = []
@@ -202,10 +247,10 @@ def main():
tires = cut_tire_at(H1, ed1, d) tires = cut_tire_at(H1, ed1, d)
candidates = [t for t in tires candidates = [t for t in tires
if t['face_length'] >= 4 if t['face_length'] >= 4
and t['inner_edges'] and t['outer_edges']] and (t['out_spokes'] or t['in_spokes'])]
if candidates: if candidates:
candidates.sort(key=lambda t: -(len(t['inner_edges']) + candidates.sort(key=lambda t: -(len(t['out_spokes']) +
len(t['outer_edges']))) len(t['in_spokes'])))
selected.append((d, candidates[0])) selected.append((d, candidates[0]))
print(f"\nCut tires selected for visualization at depths: " print(f"\nCut tires selected for visualization at depths: "
f"{[d for d, _ in selected]}") f"{[d for d, _ in selected]}")
@@ -225,18 +270,20 @@ def main():
draw_cut_tire(ax, H1, pos1, ed1, tire, d, draw_cut_tire(ax, H1, pos1, ed1, tire, d,
title=(f"$d = {d}$: face length " title=(f"$d = {d}$: face length "
f"{tire['face_length']}, " f"{tire['face_length']}, "
f"{len(tire['inner_edges'])} inner + " f"{len(tire['out_spokes'])} out + "
f"{len(tire['outer_edges'])} outer spokes")) f"{len(tire['in_spokes'])} in spokes"))
ax.set_xlim(min(xs) - 0.5, max(xs) + 0.5) ax.set_xlim(min(xs) - 0.5, max(xs) + 0.5)
ax.set_ylim(min(ys) - 0.5, max(ys) + 0.5) ax.set_ylim(min(ys) - 0.5, max(ys) + 0.5)
for idx in range(n, len(axes_flat)): for idx in range(n, len(axes_flat)):
axes_flat[idx].axis('off') axes_flat[idx].axis('off')
fig.suptitle(r"Cut tires on $G'_1$ (Holton-McKay #0, V\S half) at " fig.suptitle(r"Cut tires on $G'_1$ (Holton-McKay #0, V\S half) at "
r"several depths $d$" + "\n" r"several depths $d$" + "\n"
r"Each panel: blue face boundary at depth $d$ " r"Blue = face boundary at depth $d$ (in $H_d$). "
r"+ inner spokes (depth $d-1$, red dashed) " r"Orange-square pendants = out spokes (toward depth $d-1$). "
r"+ outer spokes (depth $d+1$, green dashed)", r"Green-square pendants = in spokes (toward depth $d+1$)." + "\n"
fontsize=11, y=1.00) r"Each spoke is a labelled pendant added at a "
r"degree-$2$ boundary vertex.",
fontsize=10, y=1.00)
plt.tight_layout() plt.tight_layout()
plt.savefig(out_fig, dpi=160, bbox_inches='tight') plt.savefig(out_fig, dpi=160, bbox_inches='tight')
plt.close() plt.close()
papers/coloring_nested_tire_graphs/notes/cut_depth_label.aux
+5 -5
View File
@@ -4,9 +4,9 @@
\@writefile{toc}{\contentsline {paragraph}{Visualization.}{2}{}\protected@file@percent } \@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}{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{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}} \providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
\newlabel{fig:cut-depth-label}{{1}{4}} \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 }}{5}{}\protected@file@percent } \@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}{5}} \newlabel{fig:cut-tire}{{2}{6}}
\gdef \@abspage@last{5} \gdef \@abspage@last{6}
papers/coloring_nested_tire_graphs/notes/cut_depth_label.fdb_latexmk
+84
View File
@@ -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 ""
"/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/jknappen/ec/tcrm1095.tfm" 1136768653 1536 02c06700a42be0f5a28664c7273f82e7 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1246382020 908 2921f8a10601f252058503cc6570e581 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm" 1136768653 1328 c834bbb027764024c09d3d2bf908b5f0 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm" 1136768653 1324 c910af8c371558dc20f2d7822f66fe64 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmex10.tfm" 1136768653 992 662f679a0b3d2d53c1b94050fdaa3f50 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmitt10.tfm" 1136768653 768 2297ad2ac26f37e67f756dad27c77d68 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi10.tfm" 1136768653 1528 abec98dbc43e172678c11b3b9031252a ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi12.tfm" 1136768653 1524 4414a8315f39513458b80dfc63bff03a ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm" 1136768653 1520 eccf95517727cb11801f4f1aee3a21b4 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr10.tfm" 1136768653 1296 45809c5a464d5f32c8f98ba97c1bb47f ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr12.tfm" 1136768653 1288 655e228510b4c2a1abe905c368440826 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr17.tfm" 1136768653 1292 296a67155bdbfc32aa9c636f21e91433 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm" 1136768653 1124 6c73e740cf17375f03eec0ee63599741 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm" 1136768653 768 1321e9409b4137d6fb428ac9dc956269 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb" 1248133631 34811 78b52f49e893bcba91bd7581cdc144c0 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1248133631 32080 340ef9bf63678554ee606688e7b5339d ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmitt10.pfb" 1248133631 26057 fad158094905eaf20f4ae3782af0c45c ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1248133631 35469 70d41d2b9ea31d5d813066df7c99281c ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb" 1248133631 32362 179c33bbf43f19adbb3825bb4e36e57a ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1248133631 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb" 1248133631 32626 4f5c1b83753b1dd3a97d1b399a005b4b ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb" 1248133631 37944 359e864bd06cde3b1cf57bb20757fb06 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb" 1248133631 31099 c85edf1dd5b9e826d67c9c7293b6786c ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb" 1248133631 34694 ad62b13721ee8eda1dcc8993c8bd7041 ""
"/usr/local/texlive/2022/texmf-dist/fonts/type1/public/cm-super/sfrm1095.pfb" 1215737283 145929 f25e56369a345c4ff583b067cd87ce8e ""
"/usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii" 1461363279 71627 94eb9990bed73c364d7f53f960cc8c5b ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty" 1644112042 7237 bdd120a32c8fdb4b433cf9ca2e7cd98a ""
"/usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty" 1572645307 1057 525c2192b5febbd8c1f662c9468335bb ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty" 1591045760 12594 0d51ac3a545aaaa555021326ff22a6cc ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty" 1359763108 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty" 1359763108 13829 94730e64147574077f8ecfea9bb69af4 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd" 1359763108 961 6518c6525a34feb5e8250ffa91731cff ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd" 1359763108 961 d02606146ba5601b5645f987c92e6193 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty" 1622667781 2222 da905dc1db75412efd2d8f67739f0596 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty" 1622667781 4173 bc0410bcccdff806d6132d3c1ef35481 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty" 1636758526 87648 07fbb6e9169e00cb2a2f40b31b2dbf3c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty" 1636758526 4128 8eea906621b6639f7ba476a472036bbe ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty" 1636758526 2444 926f379cc60fcf0c6e3fee2223b4370d ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls" 1636758526 20144 8a7de377ae7a11ee924a7499611f5a9d ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/base/size11.clo" 1636758526 8464 74db94825c407b51399ca17d9bd38a3d ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty" 1579038678 6078 f1cb470c9199e7110a27851508ed7a5c ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty" 1647548653 54291 b8e5c600d4aa37b48a740dd2a6c26163 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty" 1647548653 71241 d2cd3a1c5acef9cb31f945b93c0bb6e3 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1579991033 13886 d1306dcf79a944f6988e688c1785f9ce ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty" 1578002852 41601 9cf6c5257b1bc7af01a58859749dd37a ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg" 1465944070 1224 978390e9c2234eab29404bc21b268d1e ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def" 1601931164 19103 48d29b6e2a64cb717117ef65f107b404 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty" 1622581934 18399 7e40f80366dffb22c0e7b70517db5cb4 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty" 1636758526 7996 a8fb260d598dcaf305a7ae7b9c3e3229 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty" 1622581934 2671 4de6781a30211fe0ea4c672e4a2a8166 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty" 1636758526 4009 187ea2dc3194cd5a76cd99a8d7a6c4d0 ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def" 1644269979 29921 d0acc05a38bd4aa3af2017f0b7c137ce ""
"/usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg" 1279039959 678 4792914a8f45be57bb98413425e4c7af ""
"/usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf" 1646502317 40171 cdab547de63d26590bebb3baff566530 ""
"/usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1647878959 4410336 7d30a02e9fa9a16d7d1f8d037ba69641 ""
"/usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt" 1665017617 2826443 7e98410c533054b636c6470db83a27bc ""
"/usr/local/texlive/2022/texmf.cnf" 1647878952 577 209b46be99c9075fd74d4c0369380e8c ""
"cut_depth_label.aux" 1779825368 1721 5edc7723d8b71c6ba2b4700e3c665b88 "pdflatex"
"cut_depth_label.tex" 1779825366 10920 bbdfc135e2020fd4e775aa257f60a3f1 ""
"fig_cut_depth_label.png" 1779822462 206719 3fb2fa6c688c6e4950d3218cb0d06391 ""
"fig_cut_tire.png" 1779825334 191798 f2e9e6902813cf441014987202b8b4a3 ""
(generated)
"cut_depth_label.aux"
"cut_depth_label.log"
"cut_depth_label.pdf"
papers/coloring_nested_tire_graphs/notes/cut_depth_label.fls
+343
View File
@@ -0,0 +1,343 @@
PWD /Users/didericis/Code/math-research/papers/coloring_nested_tire_graphs/notes
INPUT /usr/local/texlive/2022/texmf.cnf
INPUT /usr/local/texlive/2022/texmf-dist/web2c/texmf.cnf
INPUT /usr/local/texlive/2022/texmf-var/web2c/pdftex/pdflatex.fmt
INPUT /Users/didericis/Code/math-research/papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex
OUTPUT cut_depth_label.log
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/article.cls
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/size11.clo
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/size11.clo
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/size11.clo
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/base/size11.clo
INPUT /usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsmath.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amstext.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsbsy.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsopn.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsmath/amsgen.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amssymb.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/amsfonts.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amscls/amsthm.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphicx.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/graphics.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/trig.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-cfg/graphics.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics-def/pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/geometry/geometry.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/ifvtex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/generic/iftex/iftex.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/booktabs/booktabs.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/caption/caption3.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/graphics/keyval.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/l3backend/l3backend-pdftex.def
INPUT ./cut_depth_label.aux
INPUT cut_depth_label.aux
INPUT cut_depth_label.aux
OUTPUT cut_depth_label.aux
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/context/base/mkii/supp-pdf.mkii
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/epstopdf-pkg/epstopdf-base.sty
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/latexconfig/epstopdf-sys.cfg
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr17.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr12.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi12.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi12.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmex10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmex10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsa.fd
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/tex/latex/amsfonts/umsb.fd
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmr12.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmmi10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmex10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt10.tfm
OUTPUT cut_depth_label.pdf
INPUT /usr/local/texlive/2022/texmf-var/fonts/map/pdftex/updmap/pdftex.map
INPUT ./fig_cut_depth_label.png
INPUT ./fig_cut_depth_label.png
INPUT fig_cut_depth_label.png
INPUT ./fig_cut_depth_label.png
INPUT ./fig_cut_depth_label.png
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/jknappen/ec/tcrm1095.tfm
INPUT /usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmitt10.tfm
INPUT ./fig_cut_tire.png
INPUT ./fig_cut_tire.png
INPUT fig_cut_tire.png
INPUT ./fig_cut_tire.png
INPUT ./fig_cut_tire.png
INPUT cut_depth_label.aux
INPUT /usr/local/texlive/2022/texmf-dist/fonts/enc/dvips/cm-super/cm-super-ts1.enc
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmitt10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmr8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy8.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb
INPUT /usr/local/texlive/2022/texmf-dist/fonts/type1/public/cm-super/sfrm1095.pfb
papers/coloring_nested_tire_graphs/notes/cut_depth_label.log
+22 -20
View File
@@ -1,4 +1,4 @@
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 15:33 This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 26 MAY 2026 15:56
entering extended mode entering extended mode
restricted \write18 enabled. restricted \write18 enabled.
%&-line parsing enabled. %&-line parsing enabled.
@@ -300,51 +300,53 @@ LaTeX Warning: Float too large for page by 659.17491pt on input line 120.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
<fig_cut_tire.png, id=19, 864.98157pt x 1087.6635pt> [2]
<fig_cut_tire.png, id=25, 959.83594pt x 1087.6635pt>
File: fig_cut_tire.png Graphic file (type png) File: fig_cut_tire.png Graphic file (type png)
<use fig_cut_tire.png> <use fig_cut_tire.png>
Package pdftex.def Info: fig_cut_tire.png used on input line 172. Package pdftex.def Info: fig_cut_tire.png used on input line 196.
(pdftex.def) Requested size: 469.75502pt x 590.68202pt. (pdftex.def) Requested size: 469.75502pt x 532.31252pt.
LaTeX Warning: `h' float specifier changed to `ht'. LaTeX Warning: `h' float specifier changed to `ht'.
[2]
Overfull \hbox (46.78696pt too wide) in paragraph at lines 196--203 Overfull \hbox (46.78696pt too wide) in paragraph at lines 223--230
\OT1/cmr/m/n/10.95 The pro-ce-dure mir-rors the $4$CT cut-and-reglue scheme (\O \OT1/cmr/m/n/10.95 The pro-ce-dure mir-rors the $4$CT cut-and-reglue scheme (\O
T1/cmtt/m/n/10.95 rainbow[]proof.tex\OT1/cmr/m/n/10.95 , \OT1/cmtt/m/n/10.95 wo T1/cmtt/m/n/10.95 rainbow[]proof.tex\OT1/cmr/m/n/10.95 , \OT1/cmtt/m/n/10.95 wo
rst[]case[]proof[]sketch.tex\OT1/cmr/m/n/10.95 , rst[]case[]proof[]sketch.tex\OT1/cmr/m/n/10.95 ,
[] []
[3] [4 <./fig_cut_depth_label.png>] [5 <./fig_cut_tire.png>] [3] [4] [5 <./fig_cut_depth_label.png>] [6 <./fig_cut_tire.png>]
(./cut_depth_label.aux) ) (./cut_depth_label.aux) )
Here is how much of TeX's memory you used: Here is how much of TeX's memory you used:
4590 strings out of 478268 4592 strings out of 478268
74486 string characters out of 5846347 74512 string characters out of 5846347
375437 words of memory out of 5000000 375437 words of memory out of 5000000
22770 multiletter control sequences out of 15000+600000 22771 multiletter control sequences out of 15000+600000
479174 words of font info for 66 fonts, out of 8000000 for 9000 479342 words of font info for 67 fonts, out of 8000000 for 9000
1141 hyphenation exceptions out of 8191 1141 hyphenation exceptions out of 8191
55i,8n,63p,656b,312s stack positions out of 10000i,1000n,20000p,200000b,200000s 55i,8n,63p,656b,312s stack positions out of 10000i,1000n,20000p,200000b,200000s
{/usr/local/texlive/2022/texmf-dist/fonts/enc/dvips/cm {/usr/local/texlive/2022/texmf-dist/fonts/enc/dvips/cm
-super/cm-super-ts1.enc}</usr/local/texlive/2022/texmf-dist/fonts/type1/public/ -super/cm-super-ts1.enc}</usr/local/texlive/2022/texmf-dist/fonts/type1/public/
amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a amsfonts/cm/cmbx10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/a
msfonts/cm/cmbx12.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am msfonts/cm/cmbx12.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
sfonts/cm/cmitt10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/am
sfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams sfonts/cm/cmmi10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/ams
fonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo fonts/cm/cmmi8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfo
nts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont nts/cm/cmr10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfont
s/cm/cmr17.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/ s/cm/cmr17.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/
cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/ cm/cmr8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/
cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c cmsy10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/c
msy6.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cms msy8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmt
y8.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmti1 i10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt
0.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/cm/cmtt10 10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/
.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/amsfonts/symbols/ms msbm10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/cm-super/sfrm
bm10.pfb></usr/local/texlive/2022/texmf-dist/fonts/type1/public/cm-super/sfrm10 1095.pfb>
95.pfb> Output written on cut_depth_label.pdf (6 pages, 509609 bytes).
Output written on cut_depth_label.pdf (5 pages, 504231 bytes).
PDF statistics: PDF statistics:
96 PDF objects out of 1000 (max. 8388607) 99 PDF objects out of 1000 (max. 8388607)
56 compressed objects within 1 object stream 58 compressed objects within 1 object stream
0 named destinations out of 1000 (max. 500000) 0 named destinations out of 1000 (max. 500000)
11 words of extra memory for PDF output out of 10000 (max. 10000000) 11 words of extra memory for PDF output out of 10000 (max. 10000000)
papers/coloring_nested_tire_graphs/notes/cut_depth_label.pdf
BIN
View File
Binary file not shown.
papers/coloring_nested_tire_graphs/notes/cut_depth_label.tex
+57 -30
View File
@@ -136,34 +136,58 @@ depth-$d$ edges). Equip $H_d$ with the planar embedding inherited
from $G'_i$. from $G'_i$.
For each face $f$ of $H_d$, the \emph{cut tire at $(d, f)$} is the 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} \begin{itemize}
\item every edge on the boundary walk of $f$ (all of depth $d$ in \item Start with the boundary walk of $f$ as the \emph{face
$H_d$), and boundary} (every edge here is of depth $d$ in $H_d$).
\item every edge of $G'_i$ with at least one endpoint on the \item For each vertex $v$ on the boundary walk of $f$ that has
boundary walk of $f$ whose depth is $d - 1$ or $d + 1$. 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 \end{itemize}
splits into \emph{inner spokes} (depth $d - 1$, pointing toward the
cut) and \emph{outer spokes} (depth $d + 1$, pointing away from the \noindent\emph{Convention.} ``Out'' means toward lower depth (toward
cut). 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} \end{definition}
\paragraph{Relation to the existing tire framework.} Under the \paragraph{Relation to the existing tire framework.} Under the
correspondence between primal level structure (paper.tex) and dual correspondence between primal level structure (\texttt{paper.tex})
depth labelling, a cut tire on $G'_i$ at $(d, f)$ corresponds to: 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} \begin{itemize}
\item face boundary at depth $d$ $\longleftrightarrow$ the tire \item face boundary at depth $d$ $\longleftrightarrow$
annular subgraph $T'_{\mathrm{ann}}$ at depth $d$ from the $T'_{\mathrm{ann}}$, the cycle (or closed walk) of annular
cut (cf.\ Def.~1.15 of paper.tex); dual vertices;
\item the cut tire itself $\longleftrightarrow$ the tire annular \item in/out spokes $\longleftrightarrow$ the leaves of $D(T)$
face connector $T'_{f'}$ (cf.\ Def.~1.16); (Def.~1.7's degree-$1$ pendants attached to the cycle);
\item inner / outer spokes $\longleftrightarrow$ the inner and \item the cut tire's planar embedding $\longleftrightarrow$
outer spokes of $T'_{f'}$ (cf.\ Def.~1.17). $D(T)$'s embedding.
\end{itemize} \end{itemize}
So the cut tire is the dual-side analogue of the The cut tire is therefore not just analogous to but \emph{is} a
``tire annular face connector,'' parametrised by depth from the cut partial tire dual (up to relabelling), with depth-from-cut playing
rather than depth from a primal level source. 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).} \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} \label{fig:cut-tire}
\end{figure} \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} \begin{itemize}
\item $d = 1$: face length $12$, $5$ inner spokes (to depth-$0$ \item $d = 1$: face length $12$, $5$ out spokes (toward depth-$0$
pendants) $+$ $4$ outer spokes. This is the outermost cut pendants) $+\ 0$ in spokes. The outermost cut tire,
tire, immediately adjacent to the cut. 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 \item $d = 2$: face length $7$ (one of two symmetric faces in
$H_2$), $4 + 3$ spokes. $H_2$), $4$ out $+\ 3$ in spokes.
\item $d = 4$: face length $8$, $2 + 5$ spokes. \item $d = 4$: face length $8$, $2$ out $+\ 1$ in.
\item $d = 5$: face length $14$, $4 + 6$ spokes. \item $d = 5$: face length $14$, $4$ out $+\ 2$ in.
\item $d = 6$: face length $12$, $7 + 1$ spokes. This is the \item $d = 6$: face length $12$, $3$ out $+\ 2$ in. Innermost cut
innermost cut tire (one face left, almost no outer spokes). tire in this chain.
\end{itemize} \end{itemize}
\section*{Connection to chain pigeonhole / 4CT reducibility} \section*{Connection to chain pigeonhole / 4CT reducibility}
papers/coloring_nested_tire_graphs/notes/fig_cut_tire.png
BIN
View File
Binary file not shown.
Side by Side Swipe Overlay

Before

Width:  |  Height:  |  Size: 223 KiB

After

Width:  |  Height:  |  Size: 227 KiB