2024/06/24 by Marcel Moralès, Morales, Marcel, Nguyễn Thị Phương Dung +1
Mathematics · #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2406.16432
openalex created_date 2024/06/24 · openalex publication_date 2024/06/24 · openalex updated_date 2026/07/28
Let G be a simple, connected non bipartite graph and let IG be the edge idealof G. In our previous work we showed that L. Lovász's theorem on ear decompositions offactor-critical graphs and the canonical decomposition of a graph given by Edmonds and Gallai are basic tools for the irreducible decomposition of IkG. In this paper we use some tools from graph theory, mainly Withney's theorem on ear decompositions of 2-edge connected graphs in order to introduce a new method to make a graph factor-critical. We can describe the set ∪_ k=1∞\rm Ass (IkG) in terms of some subsets of G. We give explicit formulas for the numbers astab(IG) and dstab(IG), which are, respectively, the smallest number k such that \rm Ass (IkG)= \rm Ass (Ik+iG) for all i≥ 0 and the smallest number k such that the maximal ideal belongs to \rm Ass(IkG). We also give very simple upper bounds for astab(IG) and dstab(IG).