Betti numbers of normal edge rings

A novel approach is introduced for computing the multi-graded Betti numbers of normal edge rings. This method is employed to delve into the edge rings of three distinct classes of simple graphs that adhere to the odd-cycle condition. These classes include compact graphs, which are devoid of even cycles and satisfy the odd-cycle condition; graphs comprised of multiple paths converging at two shared vertices; and graphs introduced in HHKO that exhibit both even and odd cycles. Explicit formulas are provided for the multi-graded Betti numbers pertaining to the edge rings of these graphs.