Regularity of 3-Path Ideals of Trees and Unicyclic Graphs

Abstract

Let G be a simple graph and I3(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/ I3(G)) is bounded below by 2 ν3(G) , where ν3(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/ I3(G)) ≤ 2 ν3(G) + 2 and provide an example that shows that the given upper bound is sharp. © 2023, The Author(s), under exclusive licence to Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia.