On the gonality of Kneser graphs

Abstract

The Kneser graphs $\text{KG}(n,k)$ are a classically studied family of graphs. One known invariant of graphs is gonality (also called divisorial gonality), which is the minimum degree of a rank 1 divisor on the graph. Using known bounds on gonality of simple, connected graphs, one may obtain that the gonality of $\text{KG}(n,k)$ is bounded above by $\binom{n-1}{k}$. In 2014, Harvey and Wood showed that the treewidth (a lower bound on gonality) for $\text{KG}(n,k)$ is $\binom{n-1}{k}-1$ for $n\geq 4k^2-3k+2$. In this paper, using scramble number, another lower bound on gonality, we improve this polynomial bound and show that the gonality of $\text{KG}(n,k)$ is exactly $\binom{n-1}{k}$ for $n\geq \frac{3k^2+k+2}{2}$, and conjecture an even stricter polynomial bound using the uniform edge scramble. We then extend our argument to the family of generalized Kneser Graphs, computing the scramble number and gonality using the same polynomial bound.