1) V1 Is there a constant C such that every finite simple graph in which each cycle has fewer chords than vertices has at most C times as many edges as vertices?
open, filed Tue Aug 25 2026 08:14:24 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Fidelity uses every simple cycle and counts precisely induced nonconsecutive cycle-vertex edges. Whole proof attacks tested minimum-degree cores, BFS layers, almost-regular expanders, random-walk closure, nested cycles, and chord-density increments. Refutation attacks tested high-girth regular graphs, blowups, subdivisions, and sparse expanders; known results prevent straightforward superlinear constructions but do not yield a linear proof. Critics checked orientation, loop exclusion, cyclic wraparound, injectivity, chord deduplication, and O(n) uniformity.
Scope. Labeled finite simple undirected graphs encoded by ordered endpoint pairs; simple cycles are injective cyclic vertex sequences; chords are induced edges between cycle vertices excluding the cycle edges; one uniform linear constant.