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Equality of ordinary and symbolic powers and the Conforti-Cornuéjols conjecture for $(n-2)$-uniform clutters

Amit Roy, Kamalesh Saha

TL;DR

The paper addresses when symbolic powers of edge ideals of clutters coincide with ordinary powers, focusing on $(n-2)$-uniform clutters. It reduces the problem to the complementary edge ideals $I_c(G)$ of associated graphs $G$, and establishes a complete equivalence between $I(\mathcal{H})^{(k)}=I(\mathcal{H})^{k}$ (for all $k$ or some $k\ge2$), the packing property, and the MFMC property, with the underlying graph $G$ restricted to a short list of small graphs $K_2$, $K_3$, $P_3$, $2K_2$, $P_4$, and $C_4$ (up to isolates). The authors provide a detailed analysis via isolated-vertex reductions and a four-vertex graph classification, and they show that this class verifies the Conforti–Cornuéjols conjecture for $(n-2)$-uniform clutters. They also compare with other known families for which the conjecture holds and discuss implications for Linear Programming duality problems.

Abstract

Let $I$ be an equigenerated squarefree monomial ideal in the polynomial ring $\mathbb{K}[x_1,\ldots,x_n]$, and let $\mathcal{H}$ be a uniform clutter on the vertex set $\{x_1,\ldots,x_n\}$ such that $I=I(\mathcal{H})$ is its edge ideal. A central and challenging problem in combinatorial commutative algebra is to classify all clutters $\mathcal{H}$ for which $I(\mathcal{H})^{(k)} = I(\mathcal{H})^{k}$ for a fixed positive integer $k$, where $I(\mathcal{H})^{(k)}$ denotes the $k^{\text{th}}$ symbolic power of $I(\mathcal{H})$. In this article, we give a complete solution to this problem for $(n-2)$-uniform clutters. Moreover, we provide a simple combinatorial classification of all $(n-2)$-uniform clutters having the packing property. As a consequence, we confirm the celebrated Conforti-Cornuéjols conjecture for $(n-2)$-uniform clutters. We also compare our results with the known families of clutters for which the conjecture is known to be true. Finally, we present an application of our results to the theory of Linear Programming duality problems.

Equality of ordinary and symbolic powers and the Conforti-Cornuéjols conjecture for $(n-2)$-uniform clutters

TL;DR

The paper addresses when symbolic powers of edge ideals of clutters coincide with ordinary powers, focusing on -uniform clutters. It reduces the problem to the complementary edge ideals of associated graphs , and establishes a complete equivalence between (for all or some ), the packing property, and the MFMC property, with the underlying graph restricted to a short list of small graphs , , , , , and (up to isolates). The authors provide a detailed analysis via isolated-vertex reductions and a four-vertex graph classification, and they show that this class verifies the Conforti–Cornuéjols conjecture for -uniform clutters. They also compare with other known families for which the conjecture holds and discuss implications for Linear Programming duality problems.

Abstract

Let be an equigenerated squarefree monomial ideal in the polynomial ring , and let be a uniform clutter on the vertex set such that is its edge ideal. A central and challenging problem in combinatorial commutative algebra is to classify all clutters for which for a fixed positive integer , where denotes the symbolic power of . In this article, we give a complete solution to this problem for -uniform clutters. Moreover, we provide a simple combinatorial classification of all -uniform clutters having the packing property. As a consequence, we confirm the celebrated Conforti-Cornuéjols conjecture for -uniform clutters. We also compare our results with the known families of clutters for which the conjecture is known to be true. Finally, we present an application of our results to the theory of Linear Programming duality problems.
Paper Structure (2 sections, 7 theorems, 22 equations, 1 figure)

This paper contains 2 sections, 7 theorems, 22 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main results

Key Result

Lemma 2.1

Let $\mathcal{H}$ and $\mathcal{H}_{y_1\cdots y_r}$ be two clutters defined as above. Then $\mathcal{H}$ satisfies the packing property if and only if $\mathcal{H}_{y_1\cdots y_r}$ satisfies the packing property.

Figures (1)

  • Figure 1: All non-isomorphic graphs on four vertices without isolated vertices.

Theorems & Definitions (19)

  • Conjecture 1.1: Conforti–Cornuéjols Conjecture
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4: Symbolic power
  • Proposition 2.5
  • proof
  • Remark 2.6
  • ...and 9 more