{"id":"hexagon/2611.00066","pdf":"https://hexagonmath.org/pdf/2611.00066v1","url":"https://hexagonmath.org/abs/2611.00066v1","tags":["msc:05C35","extremal graph theory"],"score":0,"title":"Off-Diagonal Commonality of a Shorter Even Cycle and a Longer Odd Cycle","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Taeyoung Kim"],"reviews":0,"subject":"math.CO","summary":"Let $n\\ge4$ be even and let $\\ell>n$ be odd. We determine the complete off-diagonal commonality region of the ordered pair $(C_n,C_\\ell)$: it is the half-open interval $(0,a_{n,\\ell}^{\\star}]$ cut out by the unique root $a_{n,\\ell}^{\\star}\\in(1/2,1)$ of $\\frac{a^{n-1}\\bigl(n+(\\ell-n)a\\bigr)}{\\ell}=2^{1-n}.$ The upper endpoint is witnessed by the balanced disjoint union of two cliques. For the lower bound, cycle densities are expressed as traces of graphon operators. Rank-one eigenvalue majorisation separates the Perron eigenvalue from the rest of the spectrum, while the Hilbert--Schmidt norm…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT"],"discussed":0,"published":"2026-11-04","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"all"}