SCHWEITZER 1981 Problem 2

Consider the lattice $L$ of the contradictions of a simple graph $G$ (as sets of vertex pairs) with respect to inclusion. Let $n \geq 1$ be an arbitrary integer. Show that the identity $$x \bigwedge \left( \bigvee_{i=0}^n y_i \right) = \bigvee_{j=0}^n \left( x \bigwedge \left( \bigvee_{0 \leq i \leq n, i\not=j\ } y_i \right)\right)$$ holds if and only if $G$ has no cycle of size at least $n+2$ .

A. Huhn