Add small-n ELG counting experiment (iso, rooted)
count_elgs.py enumerates triangulation iso-classes and counts Even Level Graphs (G,v) per n: iso-classes (sources up to Aut) and flag-rooted (4E/|Aut| * s, an exact integer since Aut acts freely on flags). Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>
This commit is contained in:
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"""
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Count Even Level Graphs (ELGs) for small n.
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An ELG is a pair (G, S) where G is a plane triangulation on n vertices,
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S = {v} a single source vertex, and every level subgraph L_k (vertices at
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BFS-distance k from v) is bipartite -- equivalently, every level cycle is
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even (Definition: even-level-graph, sibling paper).
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We report per n:
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- tri : # iso-classes of plane triangulations (maximal planar graphs)
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- elg_iso : # iso-classes of ELG *pairs* (G,v), i.e. valid sources counted
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up to Aut(G)
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- elg_root : # flag-rooted ELGs = sum over iso-classes of 4E/|Aut(G)| * s(G),
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E = 3n-6, s(G) = # valid sources. Aut(G) acts freely on the 4E
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flags, so each term is an exact integer -- the small-integer,
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automorphism-free count (the natural one for closed forms).
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"""
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import time
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import networkx as nx
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from networkx.algorithms.isomorphism import GraphMatcher
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from triangulation_gen import enumerate_all_triangulations
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def is_even_level_graph(G, source):
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"""Every level subgraph L_k (BFS distance k from source) is bipartite,
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equivalently every level cycle is even. Mirrors test_conjecture.py in
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the even_level_graph_generators paper."""
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levels = {v: float("inf") for v in G.nodes()}
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from collections import deque
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dq = deque()
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for s in source:
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levels[s] = 0
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dq.append(s)
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while dq:
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v = dq.popleft()
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for w in G[v]:
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if levels[w] > levels[v] + 1:
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levels[w] = levels[v] + 1
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dq.append(w)
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if any(l == float("inf") for l in levels.values()):
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return False, None # disconnected: source can't reach all vertices
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for k in range(max(levels.values()) + 1):
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L_k = G.subgraph([v for v in G.nodes() if levels[v] == k])
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if not nx.is_bipartite(L_k):
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return False, None
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return True, levels
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def automorphisms(G):
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"""All automorphisms of G as node->node dict mappings."""
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return list(GraphMatcher(G, G).isomorphisms_iter())
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def valid_sources(G):
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"""Vertices v such that (G, {v}) is an ELG."""
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return [v for v in G.nodes()
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if is_even_level_graph(G, frozenset({v}))[0]]
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def source_orbits(G, sources, autos):
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"""Number of Aut(G)-orbits among the given source vertices."""
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src = set(sources)
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seen, orbits = set(), 0
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for v in sources:
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if v in seen:
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continue
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orbits += 1
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for a in autos:
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seen.add(a[v])
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return orbits
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def count_for_n(n):
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tris = enumerate_all_triangulations(n)
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flags = 4 * (3 * n - 6) # # flags of any n-vertex triangulation, E = 3n-6
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elg_iso = 0
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elg_root = 0
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n_elg_tris = 0 # triangulations admitting at least one ELG source
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for G in tris:
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srcs = valid_sources(G)
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if not srcs:
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continue
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n_elg_tris += 1
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autos = automorphisms(G)
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aut_size = len(autos)
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elg_iso += source_orbits(G, srcs, autos)
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# Aut acts freely on flags, so flags//aut_size is exact per class.
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assert flags % aut_size == 0, (n, aut_size, flags)
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elg_root += (flags // aut_size) * len(srcs)
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return {
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"tri": len(tris),
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"tri_with_elg": n_elg_tris,
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"elg_iso": elg_iso,
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"elg_root": elg_root,
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}
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if __name__ == "__main__":
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import sys
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ns = [int(x) for x in sys.argv[1:]] or [4, 5, 6, 7, 8]
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print(f"{'n':>3} {'tri':>6} {'tri+ELG':>8} {'elg_iso':>8} "
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f"{'elg_root':>9} {'time':>6}")
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for n in ns:
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t0 = time.time()
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r = count_for_n(n)
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print(f"{n:>3} {r['tri']:>6} {r['tri_with_elg']:>8} "
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f"{r['elg_iso']:>8} {r['elg_root']:>9} "
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f"{time.time()-t0:>5.1f}s")
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@@ -1,6 +1,5 @@
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 20 MAY 2026 16:00
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This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022) (preloaded format=pdflatex 2022.10.5) 22 MAY 2026 12:20
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Reference in New Issue
Block a user