Tools for proving that graphs are not graceful
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…
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