face_monochromatic_pairs: rename check_conj_3_8_scaled → check_conj_final_scaled; add n=21-24 test
Rename the shared helper module to a number-resistant name. Update all 26 dependent scripts via sed. Add experiments/test_n_21_to_24.py — extends the empirical check beyond |V(G)| ≤ 20 to n_G ∈ [21, 24]. Checks per chord-apex+Kempe colouring: (1) h_φ constant on V(K_b)? (counterexample to Corollary 5.4) (2) h_φ constant on V(K_b) ∪ V(K_c)? (counterexample to Conj 5.1) (3) Deciding face exists? Writes results incrementally to test_n_21_to_24_results.jsonl (one JSON line per triangulation, plus n-level and grand summaries). Emits PROGRESS lines every 10 minutes (default) to stdout for live monitoring. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
This commit is contained in:
@@ -28,7 +28,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -29,7 +29,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -21,7 +21,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -17,7 +17,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -21,7 +21,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -23,7 +23,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -24,7 +24,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -28,7 +28,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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+1
-1
@@ -34,7 +34,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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+1
-1
@@ -16,7 +16,7 @@ the 4-edge-face criterion). Then we:
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partial(f_n).
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Aggregates per-n.
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Run with: sage experiments/check_conj_3_8_scaled.py
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Run with: sage experiments/check_conj_final_scaled.py
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"""
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from sage.all import Graph
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from sage.graphs.graph_generators import graphs
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@@ -7,7 +7,7 @@ Each Holton-McKay graph is itself a cubic plane graph $G'$; its dual is a
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of a hypothetical minimal counterexample and apply the reduced-dual
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construction at each of its pentagonal faces.
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Reuses the testing infrastructure from check_conj_3_8_scaled.py.
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Reuses the testing infrastructure from check_conj_final_scaled.py.
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Run with: sage experiments/check_conj_on_holton_mckay.py
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"""
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@@ -19,7 +19,7 @@ from sage.all import Graph
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -28,7 +28,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -31,7 +31,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -26,7 +26,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -20,7 +20,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -28,7 +28,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -22,7 +22,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -23,7 +23,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -18,7 +18,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -24,7 +24,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -21,7 +21,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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+1
-1
@@ -33,7 +33,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -21,7 +21,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -21,7 +21,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -24,7 +24,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -24,7 +24,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -33,7 +33,7 @@ from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_3_8_scaled import (
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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@@ -0,0 +1,252 @@
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"""Test Conjecture 5.1 (face-monochromatic-pair) on chord-apex+Kempe
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colourings of reduced duals for n in [21, 24].
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For each (G, v, i, φ) chord-apex+Kempe colouring, check:
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(1) Is h_φ constant on V(K_b) ∪ V(K_c)? (= potential counterexample
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to Conjecture 5.1)
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(2) Is h_φ constant on V(K_b) alone? (= weaker condition, would
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contradict Corollary 5.4)
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(3) Does a deciding face exist (∂f ⊆ V(K_b) ∪ V(K_c) and |f| ≢ 0
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mod 3)? (= condition for the structural proof to apply)
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The known empirical result for n ≤ 20 is 0/142,812 for (1) and (2),
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and 142,812/142,812 for (3). We extend to n = 21, 22, 23, 24.
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Results are written incrementally to results.jsonl in the cwd, so
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that partial progress is preserved if the script is interrupted.
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The script also prints PROGRESS lines (= one per triangulation
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processed, or every 10 minutes elapsed, whichever first) that can
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be tailed for live monitoring.
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Run with: sage experiments/test_n_21_to_24.py
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"""
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import json
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import os
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import sys
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import time
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from pathlib import Path
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from sage.all import Graph
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from sage.graphs.graph_generators import graphs
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HERE = os.path.dirname(os.path.abspath(__file__))
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sys.path.insert(0, HERE)
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from check_conj_final_scaled import (
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apply_reduction,
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proper_3_edge_colorings,
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matches_chord_apex_kempe,
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kempe_cycle_set,
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edge_idx,
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)
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from check_heawood_on_kempe import dual_of, heawood_numbers, vertices_of_kempe
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RESULTS_FILE = Path(HERE) / '..' / 'test_n_21_to_24_results.jsonl'
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def find_deciding_face(H, V_union):
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"""Return (face, |face|) for some deciding face, or None."""
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for f in H.faces():
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verts = {u for (u, v) in f} | {v for (u, v) in f}
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if not verts.issubset(V_union):
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continue
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L = len(f)
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if L % 3 != 0:
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return f, L
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return None
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def test_one(D):
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"""For a given dual D, iterate reductions + chord-apex+Kempe
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colourings, return a dict of counters."""
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D.is_planar(set_embedding=True)
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n_col = 0
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n_const_kb = 0
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n_const_kb_kc_union = 0
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n_no_deciding = 0
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n_counterexamples = [] # save details of any counterexample
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for face in D.faces():
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if len(face) != 5: continue
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for i_red in range(5):
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res = apply_reduction(D, face, i_red, 9999)
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if res is None: continue
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H = res['H']; named = res['named']
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H.is_planar(set_embedding=True)
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edges, colorings = proper_3_edge_colorings(H)
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cand = [c for c in colorings
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if matches_chord_apex_kempe(edges, c, named)]
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for col in cand:
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n_col += 1
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try:
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h = heawood_numbers(H, edges, col)
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except RuntimeError:
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continue
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merged_idx = edge_idx(edges, named['merged'])
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a = col[merged_idx]
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bs = [c for c in range(3) if c != a]
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kc_b = kempe_cycle_set(edges, col, merged_idx, (a, bs[0]))
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kc_c = kempe_cycle_set(edges, col, merged_idx, (a, bs[1]))
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V_b = vertices_of_kempe(edges, kc_b)
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V_c = vertices_of_kempe(edges, kc_c)
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V_union = V_b | V_c
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# Check constancy
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h_b_vals = {h[v] for v in V_b}
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h_union_vals = {h[v] for v in V_union}
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if len(h_b_vals) == 1:
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n_const_kb += 1
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n_counterexamples.append({
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'type': 'const_kb',
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'n_V_b': len(V_b),
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})
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if len(h_union_vals) == 1:
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n_const_kb_kc_union += 1
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n_counterexamples.append({
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'type': 'const_union',
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'n_V_union': len(V_union),
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})
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# Check deciding face
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dec = find_deciding_face(H, V_union)
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if dec is None:
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n_no_deciding += 1
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n_counterexamples.append({
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'type': 'no_deciding',
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'n_V_union': len(V_union),
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})
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return n_col, n_const_kb, n_const_kb_kc_union, n_no_deciding, n_counterexamples
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def write_result(record):
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"""Append a single JSONL record."""
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with open(RESULTS_FILE, 'a') as f:
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f.write(json.dumps(record) + '\n')
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f.flush()
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def main(n_min=21, n_max=24, progress_every_seconds=600):
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started_at = time.time()
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# Clear / create the results file
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print(f"Writing incremental results to {RESULTS_FILE}")
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print(f"PROGRESS lines emitted every {progress_every_seconds}s "
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f"(in addition to per-triangulation logs)")
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sys.stdout.flush()
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if RESULTS_FILE.exists():
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# Preserve old log under .bak; new run starts fresh
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bak = RESULTS_FILE.with_suffix(RESULTS_FILE.suffix + '.bak')
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RESULTS_FILE.rename(bak)
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print(f"Previous results moved to {bak}")
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RESULTS_FILE.touch()
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||||
grand_col = 0
|
||||
grand_const_kb = 0
|
||||
grand_const_union = 0
|
||||
grand_no_deciding = 0
|
||||
|
||||
last_progress = time.time()
|
||||
|
||||
for n in range(n_min, n_max + 1):
|
||||
n_start = time.time()
|
||||
print(f"\n=== n_G = {n} ===")
|
||||
sys.stdout.flush()
|
||||
try:
|
||||
triangulations_iter = graphs.triangulations(n, minimum_degree=5)
|
||||
except Exception as ex:
|
||||
print(f"n={n}: cannot enumerate ({ex})")
|
||||
continue
|
||||
n_col_n = 0
|
||||
n_const_kb_n = 0
|
||||
n_const_union_n = 0
|
||||
n_no_deciding_n = 0
|
||||
tri_idx = 0
|
||||
for G in triangulations_iter:
|
||||
G.is_planar(set_embedding=True)
|
||||
D = dual_of(G)
|
||||
tri_start = time.time()
|
||||
nc, nck, ncu, nnd, examples = test_one(D)
|
||||
tri_elapsed = time.time() - tri_start
|
||||
n_col_n += nc
|
||||
n_const_kb_n += nck
|
||||
n_const_union_n += ncu
|
||||
n_no_deciding_n += nnd
|
||||
# Write per-triangulation record
|
||||
record = {
|
||||
'n_G': n, 'tri_idx': tri_idx,
|
||||
'n_col': nc,
|
||||
'n_const_kb': nck,
|
||||
'n_const_kb_kc_union': ncu,
|
||||
'n_no_deciding': nnd,
|
||||
'tri_elapsed_s': round(tri_elapsed, 2),
|
||||
'wall_elapsed_s': round(time.time() - started_at, 2),
|
||||
'examples': examples,
|
||||
}
|
||||
write_result(record)
|
||||
# Time-based progress line
|
||||
now = time.time()
|
||||
if now - last_progress >= progress_every_seconds:
|
||||
wall_min = (now - started_at) / 60
|
||||
print(f"PROGRESS: wall {wall_min:.1f} min; "
|
||||
f"n={n} tri {tri_idx+1} done ({tri_elapsed:.1f}s); "
|
||||
f"running totals @n={n}: col={n_col_n}, "
|
||||
f"const_kb={n_const_kb_n}, "
|
||||
f"const_union={n_const_union_n}, "
|
||||
f"no_dec={n_no_deciding_n}")
|
||||
sys.stdout.flush()
|
||||
last_progress = now
|
||||
tri_idx += 1
|
||||
|
||||
n_elapsed = time.time() - n_start
|
||||
# n-level summary record
|
||||
summary = {
|
||||
'type': 'n_summary',
|
||||
'n_G': n,
|
||||
'n_tri': tri_idx,
|
||||
'n_col': n_col_n,
|
||||
'n_const_kb': n_const_kb_n,
|
||||
'n_const_kb_kc_union': n_const_union_n,
|
||||
'n_no_deciding': n_no_deciding_n,
|
||||
'n_elapsed_s': round(n_elapsed, 2),
|
||||
'wall_elapsed_s': round(time.time() - started_at, 2),
|
||||
}
|
||||
write_result(summary)
|
||||
print(f"\nn={n}: {tri_idx} triangulations, {n_col_n} colourings "
|
||||
f"[{n_elapsed:.0f}s]")
|
||||
print(f" const_kb: {n_const_kb_n}")
|
||||
print(f" const_kb_kc_union: {n_const_union_n}")
|
||||
print(f" no_deciding_face: {n_no_deciding_n}")
|
||||
sys.stdout.flush()
|
||||
|
||||
grand_col += n_col_n
|
||||
grand_const_kb += n_const_kb_n
|
||||
grand_const_union += n_const_union_n
|
||||
grand_no_deciding += n_no_deciding_n
|
||||
|
||||
total_elapsed = time.time() - started_at
|
||||
final = {
|
||||
'type': 'grand_summary',
|
||||
'n_min': n_min, 'n_max': n_max,
|
||||
'grand_col': grand_col,
|
||||
'grand_const_kb': grand_const_kb,
|
||||
'grand_const_kb_kc_union': grand_const_union,
|
||||
'grand_no_deciding': grand_no_deciding,
|
||||
'wall_elapsed_s': round(total_elapsed, 2),
|
||||
}
|
||||
write_result(final)
|
||||
print("\n" + "=" * 70)
|
||||
print(f"DONE: n in [{n_min}, {n_max}], "
|
||||
f"{grand_col} chord-apex+Kempe colourings tested "
|
||||
f"in {total_elapsed/60:.1f} min")
|
||||
print(f" const_kb: {grand_const_kb}")
|
||||
print(f" const_kb_kc_union: {grand_const_union}")
|
||||
print(f" no_deciding_face: {grand_no_deciding}")
|
||||
if (grand_const_kb == 0 and grand_const_union == 0
|
||||
and grand_no_deciding == 0):
|
||||
print("✓ Conjecture 5.1 + Deciding-face conjecture EMPIRICALLY "
|
||||
"EXTENDED to n_G ≤ {}".format(n_max))
|
||||
|
||||
|
||||
if __name__ == '__main__':
|
||||
import sys
|
||||
n_min = int(sys.argv[1]) if len(sys.argv) > 1 else 21
|
||||
n_max = int(sys.argv[2]) if len(sys.argv) > 2 else 24
|
||||
main(n_min=n_min, n_max=n_max)
|
||||
@@ -40,21 +40,50 @@
|
||||
\newlabel{lem:both-kempe-constant}{{5.3}{11}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The two cases in the proof of Lemma\nonbreakingspace 5.2\hbox {}. Vertices $v_0, v_1$ are consecutive on the $\{a, b\}$-Kempe cycle $K$, joined by an edge $e$, with the lemma's hypothesis $h_\varphi (v_0) = h_\varphi (v_1) = +1$ --- so both vertices share the clockwise colour order $(a, b, c)$. \emph {Left (Case\nonbreakingspace A):} when $\varphi (e) = a$, the colour-$b$ edge at $v_0$ lies south of $e$ (on $\partial F_R$) and the colour-$b$ edge at $v_1$ lies north of $e$ (on $\partial F_L$); the two would-be witness edges are on opposite faces, so no face of $\setbox \z@ \hbox {\mathsurround \z@ $\textstyle G$}\mathaccent "0362{G}'_{v,i}$ contains both. \emph {Right (Case\nonbreakingspace B):} when $\varphi (e) = b$, the colour-$a$ edges at $v_0, v_1$ are likewise on opposite sides of $e$. In either case the clause-$(3)$ arc of Conjecture\nonbreakingspace 5.1\hbox {} cannot be realised at $e$.}}{12}{}\protected@file@percent }
|
||||
\newlabel{fig:lemma-kempe-heawood}{{5}{12}}
|
||||
\citation{Heawood1898}
|
||||
\newlabel{cor:single-cycle-non-constancy}{{5.4}{13}}
|
||||
\newlabel{rem:heawood-empirical}{{5.5}{13}}
|
||||
\newlabel{rem:conj-3-6-empirical}{{5.6}{13}}
|
||||
\newlabel{conj:face-monochromatic-pair-strengthened}{{5.7}{14}}
|
||||
\newlabel{rem:conj-3-8-empirical}{{5.8}{14}}
|
||||
\newlabel{conj:no-two-constant-kempe-cycles}{{5.5}{13}}
|
||||
\newlabel{rem:no-two-constant-kempe-cycles-counterexample}{{5.6}{13}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Smallest counterexample to Conjecture\nonbreakingspace 5.5\hbox {}: a $C_{28}$ fullerene-style cubic plane graph (12 pentagons + 4 hexagons) with a proper $3$-edge-colouring on which $h_\varphi $ is simultaneously constant ($\equiv -1$) on the red/blue $12$-cycle and the red/green $12$-cycle, which share the colour-red edge $(0, 1)$. Light-shaded nodes are on $V(K_0) \cap V(K_1)$; medium-shaded on $V(K_0) \cup V(K_1) \setminus V(K_0) \cap V(K_1)$; grey on neither.}}{14}{}\protected@file@percent }
|
||||
\newlabel{fig:no-two-constant-kempe-counterexample}{{6}{14}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A reduction of Conjecture 5.1\hbox {} via Heawood's face-sum identity}}{14}{}\protected@file@percent }
|
||||
\newlabel{eq:heawood-face-sum}{{5.1}{14}}
|
||||
\newlabel{conj:deciding-face}{{5.7}{14}}
|
||||
\newlabel{thm:deciding-face-implies-conj-5-1}{{5.8}{14}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{}{A partial structural proof of Conjecture\nonbreakingspace 5.7\hbox {}}}{15}{}\protected@file@percent }
|
||||
\newlabel{def:flank-face}{{5.9}{15}}
|
||||
\newlabel{lem:flank-length}{{5.10}{15}}
|
||||
\newlabel{lem:flank-covering-base}{{5.11}{15}}
|
||||
\newlabel{lem:flank-covering-hex}{{5.12}{16}}
|
||||
\newlabel{thm:deciding-face-partial}{{5.13}{17}}
|
||||
\newlabel{rem:n-i-6-flank-fails}{{5.14}{17}}
|
||||
\newlabel{def:outer-face}{{5.15}{17}}
|
||||
\newlabel{lem:outer-face-length}{{5.16}{17}}
|
||||
\newlabel{lem:outer-face-covering-base}{{5.17}{17}}
|
||||
\newlabel{thm:deciding-face-partial-extended}{{5.18}{18}}
|
||||
\newlabel{rem:deciding-face-remaining-case}{{5.19}{18}}
|
||||
\newlabel{conj:gprime-pentagon-fallback}{{5.20}{18}}
|
||||
\newlabel{lem:gprime-pigeonhole}{{5.21}{18}}
|
||||
\newlabel{rem:gprime-pigeonhole-stop}{{5.22}{18}}
|
||||
\citation{AH77a}
|
||||
\citation{AHK77}
|
||||
\citation{RSST97}
|
||||
\citation{Gonthier08}
|
||||
\newlabel{rem:deciding-face-empirical}{{5.23}{20}}
|
||||
\newlabel{rem:heawood-empirical}{{5.24}{20}}
|
||||
\newlabel{rem:conj-3-6-empirical}{{5.25}{21}}
|
||||
\newlabel{conj:face-monochromatic-pair-strengthened}{{5.26}{21}}
|
||||
\newlabel{rem:conj-3-8-empirical}{{5.27}{22}}
|
||||
\newlabel{rem:implication-4ct}{{5.28}{22}}
|
||||
\bibcite{Heawood1898}{1}
|
||||
\bibcite{AH77a}{2}
|
||||
\bibcite{AHK77}{3}
|
||||
\bibcite{RSST97}{4}
|
||||
\bibcite{Gonthier08}{5}
|
||||
\newlabel{tocindent-1}{0pt}
|
||||
\newlabel{tocindent0}{12.7778pt}
|
||||
\newlabel{tocindent0}{14.69437pt}
|
||||
\newlabel{tocindent1}{17.77782pt}
|
||||
\newlabel{tocindent2}{0pt}
|
||||
\newlabel{tocindent3}{0pt}
|
||||
\newlabel{rem:implication-4ct}{{5.9}{15}}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{15}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{15}
|
||||
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{23}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{23}
|
||||
|
||||
@@ -1,9 +1,15 @@
|
||||
# Fdb version 3
|
||||
["pdflatex"] 1779636334 "/Users/didericis/Code/math-research/papers/dual_decomposition_minimal_counterexamples/paper.tex" "paper.pdf" "paper" 1779636335
|
||||
"/Users/didericis/Code/math-research/papers/dual_decomposition_minimal_counterexamples/paper.tex" 1779636334 29173 3355ad986dd2045d7c83ee9e1f4197ba ""
|
||||
["pdflatex"] 1779709018 "paper.tex" "paper.pdf" "paper" 1779709020
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmbsy5.tfm" 1246382020 1120 1e8878807317373affa7f7bba4cf2f6a ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmbsy6.tfm" 1246382020 1124 14ccf5552bc7f77ca02a8a402bea8bfb ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmbsy9.tfm" 1246382020 1112 cbc11b646ccc26599775160605aaee3a ""
|
||||
"/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 ""
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex9.tfm" 1246382020 996 a18840b13b499c08ac2de96a99eda4bc ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmmib5.tfm" 1246382020 1496 c79f6914c6d39ffb3759967363d1be79 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmmib6.tfm" 1246382020 1516 a3bf6a5e7ec4401b1f52092dfaaed242 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmmib9.tfm" 1246382020 1528 159d57adcba064aab4277245c826577d ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1246382020 916 f87d7c45f9c908e672703b83b72241a3 ""
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam5.tfm" 1246382020 924 9904cf1d39e9767e7a3622f2a125a565 ""
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1246382020 928 2dc8d444221b7a635bb58038579b861a ""
|
||||
@@ -11,33 +17,55 @@
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1246382020 940 75ac932a52f80982a9f8ea75d03a34cf ""
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 ""
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx10.tfm" 1136768653 1328 c834bbb027764024c09d3d2bf908b5f0 ""
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx5.tfm" 1136768653 1332 f817c21a1ba54560425663374f1b651a ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx6.tfm" 1136768653 1344 8a0be4fe4d376203000810ad4dc81558 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx8.tfm" 1136768653 1332 1fde11373e221473104d6cc5993f046e ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbx9.tfm" 1136768653 1328 5442e22a7072966dbaf88ca900acf3f0 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmbxti10.tfm" 1136768653 1532 9162035f4e7176612125649e348e2195 ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmcsc10.tfm" 1136768653 1300 63a6111ee6274895728663cf4b4e7e81 ""
|
||||
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|
||||
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|
||||
"/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/cmmi9.tfm" 1136768653 1524 d89e2d087a9828407a196f428428ef4a ""
|
||||
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|
||||
"/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/cmr9.tfm" 1136768653 1292 6b21b9c2c7bebb38aa2273f7ca0fb3af ""
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 ""
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||||
"/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/cmsy9.tfm" 1136768653 1116 25a7bf822c58caf309a702ef79f4afbb ""
|
||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmti10.tfm" 1136768653 1480 aa8e34af0eb6a2941b776984cf1dfdc4 ""
|
||||
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||||
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||||
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||||
"/usr/local/texlive/2022/texmf-dist/fonts/tfm/public/cm/cmtt9.tfm" 1136768653 764 c98a2af25c99b73a368cf7336e255190 ""
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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[13]
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\OT1/cmr/m/it/10 Remark \OT1/cmr/m/n/10 5.8\OT1/cmr/m/it/10 . \OT1/cmr/m/n/10 T
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he strength-ened con-jec-ture was tested on the same chord-
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[20] [21]
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[22] [23] (./paper.aux)
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Reference in New Issue
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