coloring_nested_tire_graphs: redefine cut tire bullet 2 as labelled pendants
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:
@@ -40,13 +40,15 @@ def faces_of_subgraph(H, pos):
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def cut_tire_at(G_i, edge_depth, d):
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"""Build cut tires at depth d for each face of the depth-d subgraph.
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Returns list of (face_vertices, tire_subgraph_edges_by_role) where:
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tire_subgraph_edges_by_role is a dict with keys
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'face' (boundary at depth d), 'inner' (adjacent at d-1),
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'outer' (adjacent at d+1).
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"""Build cut tires at depth d for each face of H_d.
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Per the (redefined) Definition, each cut tire is:
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- The face boundary walk of f (edges of depth d in H_d), AND
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- For each vertex v on the boundary walk with degree-2 in H_d:
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a single pendant (fresh vertex n_v + fresh edge {v, n_v})
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labelled 'out spoke' if v has an incident depth-(d-1) edge
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in G'_i, or 'in spoke' if v has an incident depth-(d+1) edge.
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"""
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# Build H_d
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H_d_edges = [e for e, ed in edge_depth.items() if ed == d]
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H_d = Graph([(u, v) for (u, v) in H_d_edges],
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multiedges=False, loops=False)
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@@ -56,68 +58,124 @@ def cut_tire_at(G_i, edge_depth, d):
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return []
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faces_h_d = H_d.faces()
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results = []
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# For each vertex in H_d, compute its degree in H_d and find its
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# incident non-H_d edges in G'_i (with their depths).
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h_d_degree = {v: H_d.degree(v) for v in H_d.vertices()}
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incident_other = {v: [] for v in H_d.vertices()}
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for e, ed in edge_depth.items():
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if ed == d:
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continue
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u, w = e
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if u in incident_other:
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incident_other[u].append((w, ed))
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if w in incident_other:
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incident_other[w].append((u, ed))
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for face in faces_h_d:
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# face is a list of (u, v) edge-pairs going around the boundary
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# in Sage; convert to vertex sequence
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vertices_on_face = []
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for (u, v) in face:
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vertices_on_face.append(u)
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face_vertex_set = set(vertices_on_face)
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# Boundary edges of the face (in H_d): they're the face edges
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vertices_on_face = [u for (u, v) in face]
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face_vertex_list = list(dict.fromkeys(vertices_on_face)) # ordered, unique
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face_edges = [(min(u, v), max(u, v)) for (u, v) in face]
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# Find adjacent edges at d-1 and d+1 from G'_i
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inner_edges = [] # depth d-1
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outer_edges = [] # depth d+1
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for e_full, ed in edge_depth.items():
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if ed not in (d - 1, d + 1):
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# For each degree-2 boundary vertex, add a labelled pendant
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out_spokes = [] # list of (boundary_v, pendant_v)
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in_spokes = [] # list of (boundary_v, pendant_v)
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pendant_to_boundary = {}
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for v in face_vertex_list:
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if h_d_degree[v] != 2:
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continue
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u, v = e_full
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if u in face_vertex_set or v in face_vertex_set:
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# In a cubic G'_i, v has exactly one non-H_d edge.
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others = incident_other.get(v, [])
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if not others:
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continue
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# Look at the depth of the unique non-H_d incident edge.
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for (w, ed) in others:
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if ed == d - 1:
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inner_edges.append(e_full)
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else:
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outer_edges.append(e_full)
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pendant_id = ('out', v)
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pendant_to_boundary[pendant_id] = v
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out_spokes.append((v, pendant_id))
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elif ed == d + 1:
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pendant_id = ('in', v)
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pendant_to_boundary[pendant_id] = v
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in_spokes.append((v, pendant_id))
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results.append({
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'face_vertices': vertices_on_face,
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'face_vertices_unique': face_vertex_list,
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'face_edges': face_edges,
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'inner_edges': inner_edges,
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'outer_edges': outer_edges,
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'out_spokes': out_spokes, # (boundary_v, pendant_id) for d-1
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'in_spokes': in_spokes, # (boundary_v, pendant_id) for d+1
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'face_length': len(face),
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})
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return results
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def draw_cut_tire(ax, G_i, pos, edge_depth, tire, d, title):
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"""Draw G_i faded with a single cut tire highlighted."""
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"""Draw G_i faded with a single cut tire highlighted.
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The cut tire (under the redefinition) is:
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- face boundary edges (depth d in H_d), and
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- pendant edges from degree-2 boundary vertices: 'out spokes'
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for those incident to a depth-(d-1) edge in G'_i, 'in spokes'
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for those incident to a depth-(d+1) edge.
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"""
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face_set = set(tire['face_edges'])
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inner_set = set(tire['inner_edges'])
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outer_set = set(tire['outer_edges'])
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face_vertex_set = set(tire['face_vertices'])
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face_vertex_set = set(tire['face_vertices_unique'])
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out_spokes = tire['out_spokes']
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in_spokes = tire['in_spokes']
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# Fade all G_i edges
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for e in G_i.edges(labels=False):
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u, v = e
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e_norm = (min(u, v), max(u, v))
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if e_norm in face_set or e_norm in inner_set or e_norm in outer_set:
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if e_norm in face_set:
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continue
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x1, y1 = pos[u]; x2, y2 = pos[v]
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ax.plot([x1, x2], [y1, y2], color='#dddddd', linewidth=0.9,
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zorder=1)
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# Highlight the tire
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# Highlight the face boundary
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for e in face_set:
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u, v = e
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x1, y1 = pos[u]; x2, y2 = pos[v]
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ax.plot([x1, x2], [y1, y2], color='#1f77b4', linewidth=3.0,
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zorder=3, label=None)
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for e in inner_set:
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u, v = e
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x1, y1 = pos[u]; x2, y2 = pos[v]
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zorder=3)
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# Compute pendant positions: each pendant vertex sits offset from
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# its boundary vertex in the direction of its parent-graph
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# incident edge (so the pendant visually replaces that edge).
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incident_other = {v: [] for v in G_i.vertices()}
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for e, ed in edge_depth.items():
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if ed == d:
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continue
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u, w = e
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if u in incident_other:
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incident_other[u].append((w, ed))
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if w in incident_other:
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incident_other[w].append((u, ed))
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pendant_pos = {}
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OFFSET = 0.20
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for (v, pid) in out_spokes + in_spokes:
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# Direction: toward the parent-graph neighbor at depth d-1 or d+1
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target_depth = d - 1 if pid[0] == 'out' else d + 1
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target = None
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for (w, ed) in incident_other.get(v, []):
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if ed == target_depth:
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target = w
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break
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if target is not None and target in pos:
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tx, ty = pos[target]; vx, vy = pos[v]
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dx, dy = tx - vx, ty - vy
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norm = (dx*dx + dy*dy) ** 0.5 or 1.0
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pendant_pos[pid] = (vx + OFFSET * dx / norm,
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vy + OFFSET * dy / norm)
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else:
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vx, vy = pos[v]
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pendant_pos[pid] = (vx + OFFSET, vy)
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# Draw spokes
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for (v, pid) in out_spokes:
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x1, y1 = pos[v]; x2, y2 = pendant_pos[pid]
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ax.plot([x1, x2], [y1, y2], color='#d62728', linewidth=2.4,
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linestyle='--', zorder=2)
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for e in outer_set:
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u, v = e
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x1, y1 = pos[u]; x2, y2 = pos[v]
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for (v, pid) in in_spokes:
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x1, y1 = pos[v]; x2, y2 = pendant_pos[pid]
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ax.plot([x1, x2], [y1, y2], color='#2ca02c', linewidth=2.4,
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linestyle='--', zorder=2)
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@@ -130,14 +188,25 @@ def draw_cut_tire(ax, G_i, pos, edge_depth, tire, d, title):
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markeredgewidth=1.0)
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else:
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ax.plot(x, y, 'o', color='#bbbbbb', markersize=5, zorder=2)
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# Draw pendant vertices
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for (v, pid) in out_spokes:
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px, py = pendant_pos[pid]
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ax.plot(px, py, 's', color='#ffaa66', markersize=8, zorder=4,
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markeredgecolor='#aa5500')
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for (v, pid) in in_spokes:
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px, py = pendant_pos[pid]
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ax.plot(px, py, 's', color='#a0e0a0', markersize=8, zorder=4,
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markeredgecolor='#2a7a2a')
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legend = [
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plt.Line2D([], [], color='#1f77b4', linewidth=3,
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label=f'face boundary (depth {d})'),
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plt.Line2D([], [], color='#d62728', linewidth=2.4,
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linestyle='--', label=f'inner spokes (depth {d - 1})'),
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linestyle='--',
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label=f'out spokes (toward depth {d - 1})'),
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plt.Line2D([], [], color='#2ca02c', linewidth=2.4,
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linestyle='--', label=f'outer spokes (depth {d + 1})'),
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linestyle='--',
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label=f'in spokes (toward depth {d + 1})'),
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]
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ax.legend(handles=legend, loc='upper left',
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bbox_to_anchor=(1.02, 1.0), fontsize=9, frameon=False)
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@@ -164,37 +233,13 @@ def main():
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# Try several depths in G'_1 (which has depths 0-7) and pick a
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# decent example.
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print(f"G'_1 depths range: 0 to {max(ed1.values())}")
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chosen_d = None
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chosen_tire = None
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for d in range(1, max(ed1.values())):
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tires = cut_tire_at(H1, ed1, d)
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print(f' depth {d}: {len(tires)} faces in depth-{d} subgraph')
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for tire in tires:
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print(f' face length {tire["face_length"]}, '
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f'inner spokes: {len(tire["inner_edges"])}, '
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f'outer spokes: {len(tire["outer_edges"])}')
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# Pick the largest "interesting" face: try to find one with
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# > 3 face edges AND inner/outer spokes both > 0.
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candidates = [t for t in tires
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if t['face_length'] >= 4
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and t['inner_edges'] and t['outer_edges']]
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if candidates and chosen_tire is None:
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chosen_d = d
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# Pick the face with the most spokes total (most "informative")
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candidates.sort(key=lambda t: -(len(t['inner_edges']) +
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len(t['outer_edges'])))
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chosen_tire = candidates[0]
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if chosen_tire is None:
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print('No suitable cut tire found.')
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return
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print(f"\nChosen cut tire at depth d = {chosen_d}")
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print(f" face length: {chosen_tire['face_length']}")
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print(f" inner spokes (depth {chosen_d - 1}): "
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f"{len(chosen_tire['inner_edges'])}")
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print(f" outer spokes (depth {chosen_d + 1}): "
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f"{len(chosen_tire['outer_edges'])}")
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print(f' depth {d}: {len(tires)} faces in H_d')
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for t in tires:
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print(f' face length {t["face_length"]}, '
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f'out spokes: {len(t["out_spokes"])}, '
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f'in spokes: {len(t["in_spokes"])}')
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# Collect "best" cut tires at multiple depths
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selected = []
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@@ -202,10 +247,10 @@ def main():
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tires = cut_tire_at(H1, ed1, d)
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candidates = [t for t in tires
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if t['face_length'] >= 4
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and t['inner_edges'] and t['outer_edges']]
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and (t['out_spokes'] or t['in_spokes'])]
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if candidates:
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candidates.sort(key=lambda t: -(len(t['inner_edges']) +
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len(t['outer_edges'])))
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candidates.sort(key=lambda t: -(len(t['out_spokes']) +
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len(t['in_spokes'])))
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selected.append((d, candidates[0]))
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print(f"\nCut tires selected for visualization at depths: "
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f"{[d for d, _ in selected]}")
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@@ -225,18 +270,20 @@ def main():
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draw_cut_tire(ax, H1, pos1, ed1, tire, d,
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title=(f"$d = {d}$: face length "
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f"{tire['face_length']}, "
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f"{len(tire['inner_edges'])} inner + "
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f"{len(tire['outer_edges'])} outer spokes"))
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f"{len(tire['out_spokes'])} out + "
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f"{len(tire['in_spokes'])} in spokes"))
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ax.set_xlim(min(xs) - 0.5, max(xs) + 0.5)
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ax.set_ylim(min(ys) - 0.5, max(ys) + 0.5)
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for idx in range(n, len(axes_flat)):
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axes_flat[idx].axis('off')
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fig.suptitle(r"Cut tires on $G'_1$ (Holton-McKay #0, V\S half) at "
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r"several depths $d$" + "\n"
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r"Each panel: blue face boundary at depth $d$ "
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r"+ inner spokes (depth $d-1$, red dashed) "
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r"+ outer spokes (depth $d+1$, green dashed)",
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fontsize=11, y=1.00)
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r"Blue = face boundary at depth $d$ (in $H_d$). "
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r"Orange-square pendants = out spokes (toward depth $d-1$). "
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r"Green-square pendants = in spokes (toward depth $d+1$)." + "\n"
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r"Each spoke is a labelled pendant added at a "
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r"degree-$2$ boundary vertex.",
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fontsize=10, y=1.00)
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plt.tight_layout()
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plt.savefig(out_fig, dpi=160, bbox_inches='tight')
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plt.close()
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@@ -4,9 +4,9 @@
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\@writefile{toc}{\contentsline {paragraph}{Visualization.}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{Relation to the existing tire framework.}{2}{}\protected@file@percent }
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\@writefile{toc}{\contentsline {paragraph}{Example on $G'_1$ (Holton-McKay \#0, $V \setminus S$ half).}{3}{}\protected@file@percent }
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\@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 }
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\@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 }
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\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
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\newlabel{fig:cut-depth-label}{{1}{4}}
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\@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 }
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\newlabel{fig:cut-tire}{{2}{5}}
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\gdef \@abspage@last{5}
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\newlabel{fig:cut-depth-label}{{1}{5}}
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\@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 }
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\newlabel{fig:cut-tire}{{2}{6}}
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\gdef \@abspage@last{6}
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@@ -0,0 +1,84 @@
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# Fdb version 3
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INPUT ./fig_cut_tire.png
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INPUT ./fig_cut_tire.png
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INPUT fig_cut_tire.png
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INPUT ./fig_cut_tire.png
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INPUT ./fig_cut_tire.png
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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
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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
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restricted \write18 enabled.
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File: fig_cut_tire.png Graphic file (type png)
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Package pdftex.def Info: fig_cut_tire.png used on input line 172.
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\OT1/cmr/m/n/10.95 The pro-ce-dure mir-rors the $4$CT cut-and-reglue scheme (\O
|
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T1/cmtt/m/n/10.95 rainbow[]proof.tex\OT1/cmr/m/n/10.95 , \OT1/cmtt/m/n/10.95 wo
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rst[]case[]proof[]sketch.tex\OT1/cmr/m/n/10.95 ,
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[]
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[3] [4 <./fig_cut_depth_label.png>] [5 <./fig_cut_tire.png>]
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[3] [4] [5 <./fig_cut_depth_label.png>] [6 <./fig_cut_tire.png>]
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Binary file not shown.
@@ -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}
|
||||
|
||||
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Reference in New Issue
Block a user