Gluing Associated Prime Ideals of Small Height

Let $B$ be a local (Noetherian) ring and suppose that $B$ has $n$ associated prime ideals where $n \geq 2$. We identify sufficient conditions for there to exist a local (Noetherian) subring $S$ of $B$ such that $S$ and $B$ have the same completion and $S$ has exactly $n - 1$ associated prime ideals. We include applications and consequences of this result.Comment: 14 page

Bounds on higher graph gonality

We prove new lower and upper bounds on the higher gonalities of finite graphs. These bounds are generalizations of known upper and lower bounds for first gonality to higher gonalities, including upper bounds on gonality involving independence number, and lower bounds on gonality by scramble number. We apply our bounds to study the computational complexity of computing higher gonalities, proving that it is NP-hard to compute the second gonality of a graph when restricting to multiplicity-free divisors.Comment: 14 pages, 5 figure