Regularity of $3$-Path Ideals of Trees and Unicyclic Graphs

TL;DR

This work analyzes the Castelnuovo–Mumford regularity of the $3$-path ideal $I_3(G)$ of a graph $G$ and ties it to the graph invariant $\nu_3(G)$, the $3$-path induced matching number, establishing a universal lower bound $\operatorname{reg}(R/I_3(G)) \geq 2\nu_3(G)$ and showing its sharpness. It proves the exact regularity for trees, $\operatorname{reg}(R/I_3(G))=2\nu_3(G)$, using inductive arguments and short exact sequences, and provides a sharp upper bound for unicyclic graphs, $\operatorname{reg}(R/I_3(G)) \leq 2\nu_3(G)+2$, with examples demonstrating the bound’s tightness. The approach relies on algebraic tools such as algebra retracts, complete intersections on induced subgraphs, and Peeva–type analyses of short exact sequences to relate subgraph structure to regularity. The results advance the understanding of how combinatorial graph properties govern algebraic invariants of path ideals and identify precise classifications where the lower or near-lower bounds are attained.

Abstract

Let $G$ be a simple graph and $I_3(G)$ be its $3$-path ideal in the corresponding polynomial ring $R$. In this article, we prove that for an arbitrary graph $G$, $reg(R/I_3(G))$ is bounded below by $2ν_3(G)$, where $ν_3(G)$ denotes the $3$-path induced matching number of $G$. We give a class of graphs, namely, trees for which the lower bound is attained. Also, for a unicyclic graph $G$, we show that $reg(R/I_3(G))\leq 2ν_3(G)+2$ and provide an example that shows that the given upper bound is sharp.

Figures (2)

• Figure 1: $G$
• Figure 2: Unicyclic graphs

Theorems & Definitions (17)

• Definition 2.1
• Remark 2.2
• Example 2.3
• Proposition 3.1
• proof
• Corollary 3.2
• proof
• Lemma 4.1
• proof
• Remark 4.2