Abstract
ABSTRACT For a graph the edge ring is , where and I(G) is generated by . The conjecture we treat is the following. Conjecture 1. If has a 2-linear resolution, then the projective dimension of , pd , 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 [7, Conjecture 4.13], and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq [1]. That there are counterexamples was noted already by Moradi and Kiani [9]. By interpreting as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds.
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