Krull dimension and regularity of binomial edge ideals of block graphs.

*(English)*Zbl 1444.13029Let \(K\) be a field, \(S=K[x_i,y_j]_{1\leq i,j\leq n}\) and \(G\) be a simple undirected graph with vertex set \([n]\). The
ideal \(J_G\) of \(S\) generated by all \(x_iy_j-x_jy_i\) such that \(i< j\) and \(\{i,j\}\) is an edge of \(G\) is called the
binomial edge ideal of \(G\). Also a block graph means a graph, in which, all blocks are cliques. In this paper, the
authors present linear time algorithms for computing the Krull dimension and the Castelnuovo-Mumford regularity of
\(S/J_G\), when \(G\) is a block graph. They also give a lower bound on the regularity of binomial edge ideals of block
graphs.

For finding the Krull dimension of \(S/J_G\), they use a result of the paper [J. Herzog et al., Adv. Appl. Math. 45, No. 3, 317–333 (2010; Zbl 1196.13018)] which characterizes minimal prime ideals of \(J_G\). Here the authors present an algorithm to find a minimal prime ideal of \(J_G\) with the minimum possible height with just one traverse of \(G\).

To study the regularity of \(S/J_G\), the authors introduce flower graphs. They call a graph a flower graph, when it is obtained by unifying a vertex \(v\) of some disjoint copies of a triangle and some disjoint copies of \(K_{1,3}\), where \(v\) must have degree 1 in each copy of \(K_{1,3}\) and the total number of copies of the triangle and \(K_{1,3}\) must be at least 3. They find extremal Betti numbers of \(S/J_G\) and state a formula for the regularity of \(S/J_G\) when \(G\) is a flower graph. Also if \(G\) is a block graph which is flower-free, that is, \(G\) has no induced subgraph which is a flower graph, a formula for the regularity of \(S/J_G\) is presented. Moreover, they use the results on flower graphs to state a lower bound on the regularity of \(S/J_G\) for general block graphs. Finally, they present and prove the correctness of an algorithm which computes the regularity of \(S/J_G\), by successively deleting certain flower graphs from \(G\) and summing up the regularities of the remained connected components which are flower-free graphs.

For finding the Krull dimension of \(S/J_G\), they use a result of the paper [J. Herzog et al., Adv. Appl. Math. 45, No. 3, 317–333 (2010; Zbl 1196.13018)] which characterizes minimal prime ideals of \(J_G\). Here the authors present an algorithm to find a minimal prime ideal of \(J_G\) with the minimum possible height with just one traverse of \(G\).

To study the regularity of \(S/J_G\), the authors introduce flower graphs. They call a graph a flower graph, when it is obtained by unifying a vertex \(v\) of some disjoint copies of a triangle and some disjoint copies of \(K_{1,3}\), where \(v\) must have degree 1 in each copy of \(K_{1,3}\) and the total number of copies of the triangle and \(K_{1,3}\) must be at least 3. They find extremal Betti numbers of \(S/J_G\) and state a formula for the regularity of \(S/J_G\) when \(G\) is a flower graph. Also if \(G\) is a block graph which is flower-free, that is, \(G\) has no induced subgraph which is a flower graph, a formula for the regularity of \(S/J_G\) is presented. Moreover, they use the results on flower graphs to state a lower bound on the regularity of \(S/J_G\) for general block graphs. Finally, they present and prove the correctness of an algorithm which computes the regularity of \(S/J_G\), by successively deleting certain flower graphs from \(G\) and summing up the regularities of the remained connected components which are flower-free graphs.

Reviewer: A. Nikseresht (Shiraz)

##### MSC:

13F65 | Commutative rings defined by binomial ideals, toric rings, etc. |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

13P99 | Computational aspects and applications of commutative rings |

13D02 | Syzygies, resolutions, complexes and commutative rings |

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\textit{C. Mascia} and \textit{G. Rinaldo}, J. Algebra Appl. 19, No. 7, Article ID 2050133, 17 p. (2020; Zbl 1444.13029)

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