  For a graph $G=(V,E)$ the edge ring $k[G]$ is $k[x_1,\ldots,x_n]/I(G)$, where
$n=|V|$ and $I(G)$ is generated by $\{ x_ix_j;\{ i,j\}\in E\}$. The conjecture
we treat is the following.
  If $k[G]$ has a 2-linear resolution, then the projective dimension of $K[G]$,
pd$(k[G])$, equals the maximal degree of a vertex in $G$.
  As far as we know, this conjecture is first mentioned in a paper by Gitler
and Valencia, and there it is called the Eliahou-Villarreal conjecture. The
conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq. That there
are counterexamples was noted already by Moradi and Kiani. By interpreting
$k[G]$ as a Stanley-Reisner ring, we are able to characterize those graphs for
which the conjecture holds.
