Initially Regular Sequences on Cycles and Depth of Unicyclic Graphs

In this article, we establish initially regular sequences on cycles of the form $C_{3n+2}$ for $n\ge 1$, in the sense of \cite{FHM-ini}. These sequences accurately compute the depth of these cycles, completing the case of finding effective initially regular sequences on cycles. Our approach involves a careful analysis of associated primes of initial ideals of the form ${\mathrm{i}\mathrm{n}\mathrm{i}}_{>}(I,f)$ for arbitrary monomial ideals $I$ and $f$ linear sums. We describe the minimal associated primes of these ideals in terms of the minimal primes of $I$. Moreover, we obtain a description of the embedded associated primes of arbitrary monomial ideals. Finally, we accurately compute the depth of certain types of unicyclic graphs.