{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:35:11Z","timestamp":1776864911287,"version":"3.51.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Let $I\\subset S=K[x_1,\\dots,x_n]$ be a squarefree monomial ideal, $K$ a field. The $k$th squarefree power $I^{[k]}$ of $I$ is the monomial ideal of $S$ generated by all squarefree monomials belonging to $I^k$. The biggest integer $k$ such that $I^{[k]}\\ne(0)$ is called the monomial grade of $I$ and it is denoted by $\\nu(I)$. Let $d_k$ be the minimum degree of the monomials belonging to $I^{[k]}$. Then, $\\text{depth}(S\/I^{[k]})\\ge d_k-1$ for all $1\\le k\\le\\nu(I)$. The normalized depth function of $I$ is defined as $g_{I}(k)=\\text{depth}(S\/I^{[k]})-(d_k-1)$, $1\\le k\\le\\nu(I)$. It is expected that $g_I(k)$ is a non-increasing function for any $I$. In this article we study the behaviour of $g_{I}(k)$ under various operations on monomial ideals. Our main result characterizes all cochordal graphs $G$ such that for the edge ideal $I(G)$ of $G$ we have $g_{I(G)}(1)=0$. They are precisely all cochordal graphs $G$ whose complementary graph $G^c$ is connected and has a cut vertex. As a far-reaching application, for given integers $1\\le s&lt;m$ we construct a graph $G$ such that $\\nu(I(G))=m$ and $g_{I(G)}(k)=0$ if and only if $k=s+1,\\dots,m$. Finally, we show that any non-increasing function of non-negative integers is the normalized depth function of some squarefree monomial ideal.<\/jats:p>","DOI":"10.37236\/11611","type":"journal-article","created":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T07:19:52Z","timestamp":1685690392000},"source":"Crossref","is-referenced-by-count":8,"title":["Behaviour of the Normalized Depth Function"],"prefix":"10.37236","volume":"30","author":[{"given":"Antonino","family":"Ficarra","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00fcrgen","family":"Herzog","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Takayuki","family":"Hibi","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2023,6,2]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p31\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i2p31\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,2]],"date-time":"2023-06-02T07:19:53Z","timestamp":1685690393000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i2p31"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,2]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,4,7]]}},"URL":"https:\/\/doi.org\/10.37236\/11611","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,2]]},"article-number":"P2.31"}}