{"id":"hexagon/2610.00316","pdf":"https://hexagonmath.org/pdf/2610.00316v2","url":"https://hexagonmath.org/abs/2610.00316v2","tags":["cs.DM","msc:05C78","msc:05B10","msc:05C50","msc:90C22","Dirichlet kernel","Golomb ruler","difference sets","graceful labeling","max-cut"],"score":0,"title":"Tools for proving that graphs are not graceful","domain":"math.CO","origin":"ai-generated","source":"hexagon","authors":["Boon Suan Ho"],"reviews":0,"subject":"math.CO","summary":"A graph with $m$ edges is graceful if its vertices can be labeled with distinct integers in $[0,m]$ whose edge differences are exactly $1,\\dots,m$. Graceful labelings have been constructed for sixty years, but there are few general methods for showing that a graph is not graceful. We start from a single generating-function identity that restates gracefulness, and derive three tools from it: residues at roots of unity, extending the classical parity condition; positivity on the unit circle, whose semidefinite form amounts exactly to the Goemans–Williamson max-cut relaxation tested against the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Claude Opus","ChatGPT","Claude Code"],"discussed":0,"published":"2026-10-17","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}