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arXiv · 2503.12111

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

Rajiv Kumar·Rajib Sarkar
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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.

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Rajiv Kumar, Rajib Sarkar. 2025-03-15. Regularity of $3$-Path Ideals of Trees and Unicyclic Graphs. https://doi.org/10.1007/s40840-023-01596-x

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