Comparing Powers of Edge Ideals

Symbolic powers of edge ideals of graphs

“…Our next lemma extends that of [23,Corollaries 5.3 and 5.4] to any unicyclic graphs. This lemma, in fact, can be derived back by a careful analysis of the proof of Theorem 3.4 and structures of indecomposable graphs.…”

“…Theorem 3.4 allows us to quickly recover and extend results of [23] on containments between symbolic and ordinary powers, on the Waldschmidt constant and the resurgence number (see Theorem 3.6).…”

“…We shall show that most of main results of [23] in fact follow from a rather nice decomposition of symbolic powers of edge ideals of unicyclic graphs. We shall also give nontrivial supportive evidence for the conjectured equality (1.1) by computing explicitly the regularity of all symbolic powers of I(G) when G is an odd cycle.…”

“…Let k = |E(T )| be the number of edges in this forest. We shall use induction on k. For k = 0, the conclusion is that of [23,Corollaries 5.3 and 5.4]. Suppose that k ≥ 1.…”

“…Our work is inspired by a recent preprint of Janssen, Kamp and Vander Woude [23], which investigates symbolic powers of the edge ideal of an odd cycle, and a general conjecture by N.C. Minh (see, for example, [8,34]), which states that for the edge ideal I = I(G) of any graph and any s ∈ N, reg I (s) = reg I s .…”