  Three graphs related to the \EGC\, are presented. The graphs are derived from
the Buckyball, the Petersen graph, and the Tutte-Coxeter graph. The first graph
is a partial answer to a question posed by Heckman and Krakovski \cite{planar}
in their recent work on the planar version of the conjecture. The other two
graphs appear to be the smallest known cubic graphs with no $2^m$-cycles for $m
\leq 4$ and for $m \leq 5$.
