Solution to a conjecture on edge rings with 2-linear resolutions

For a graph G=(V,E) the edge ring k[G] is k[x1,…,xn]/I(G), where n=|V| and I(G) generated by {xixj;{i,j}∈E}. The conjecture we treat following.Conjecture 1. If has 2-linear resolution, then projective dimension of K[G], pd (k[G]), equals maximal degree vertex in G.As far as know, this first mentioned paper Gitler Valencia [7, Conjecture 4.13], there it called Eliahou-Villarreal conjecture. treated recent Ahmed, Mafi, Namiq [Citation1]. That are counterexamples was noted already Moradi Kiani [Citation9]. By interpreting Stanley-Reisner ring, able to characterize those graphs for which holds.