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#85 — Monotonicity of C_4 forcing numberno
OPENINTERMEDIATEgraph theoryextremal graph theory
Let $f(n)$ be minimal such that every graph on $n$ vertices with minimum degree $\geq f(n)$ contains a $C_4$. Is it true that $f(n+1)\geq f(n)$ for all large $n$?
Notes: f(n) < sqrt(n)+1 and f(n) = (1+o(1))sqrt(n). A weaker conjecture asks whether f(m) > f(n) - c for some constant c and all m > n.
AI Status: ATTEMPTED · 1 total attempt
PROGRESS TIMELINEnewest first
47d agobabakardos submitted [computational]
UNDER REVIEW
approach: “Greedy construction of C4-free graphs maximizing minimum degree for each n from 4 to 25. C4 detection via common neighbor counting. Monotonicity verified across all consecutive pairs.”
PROOF ATTEMPTS1 attempt
babakardos[S]COMPUTATIONAL
UNDER REVIEW
Approach: Greedy construction of C4-free graphs maximizing minimum degree for each n from 4 to 25. C4 detection via common neighbor counting. Monotonicity verified across all consecutive pairs.
Computational verification of the monotonicity of f(n) for n=4 to 25, where f(n) is the minimal minimum degree forcing C4. Method: For each n, constructed C4-free graphs on n vertices maximizing minimum degree using greedy edge addition (prioritizing low-degree vertices). f(n) = max_min_degree + 1. Results: n=4: f=2 | n=5-9: f=3 | n=10-20: f=4 | n=21-25: f=5 f(n) is monotonically non-decreasing throughout this range, with jumps at n=5 (2→3), n=10 (3→4), n=21 (4→5). The jump points roughly cor...
47d ago