2023/11/04 by Deshpande, Priyavrat, Roy, Amit, Singh, Anurag +1 · 4 citations
#05E45 #13F55 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.2311.02430
A celebrated theorem of Fröberg gives a complete combinatorial classification of quadratic square-free monomial ideals with a linear resolution. A generalization of this theorem to higher degree square-free monomial ideals is an active area of research. The existence of a linear resolution of such ideals often depends on the field over which the polynomial ring is defined. Hence, it is too much to expect that in the higher degree case a linear resolution can be identified purely using a combinatorial feature of an associated combinatorial structure. However, some classes of ideals having linear resolutions have been identified using combinatorial structures. In the present paper, we use the notion of r-independence to construct an r-uniform hypergraph from the given graph. We then show that when the underlying graph is co-chordal, the corresponding edge ideal is vertex splittable, a condition stronger than having a linear resolution. We use this result to explicitly compute graded Betti numbers for various graph classes. Finally, we give a different proof for the existence of a linear resolution using the topological notion of r-collapsibility.