Betti numbers of normal edge rings (\bf{I})

We introduce a novel approach named the {\it induced-subgraph approach} for investigating the Betti numbers of normal edge rings. Utilizing this approach, we compute all the multi-graded Betti numbers of the edge rings associated with two-ear graphs (Definition 4.1) and compact graphs (Definition 5.1). In particular, we show that for two-ear graphs and compact graphs of type 1 or 2, their multi-graded Betti numbers are always equal to the top multi-graded Betti numbers of some of their induced subgraphs. In contrast, some of the multi-graded Betti numbers of compact graphs of type 3 are not the top multi-graded Betti numbers of any of their induced subgraph. We speculate that our approach can be applicable to many other normal edge rings.