发表机构
California State University, Fullerton; Fort Valley State University; University of Texas at Tyler; University of Kentucky（富尔顿加州州立大学; 福尔特谷州立大学; 泰勒德州大学; 肯塔基大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用scramble数改进克内泽尔图亏格的多项式界，证明特定条件下克内泽尔图亏格等于对应二项式系数，并将结论推广到广义克内泽尔图。
AI 中文摘要
克内泽尔图$\text{KG}(n,k)$是一类经典研究的图族。图的一个已知不变量是亏格（也叫除子亏格），它是图上秩1除子的最小次数。利用简单连通图亏格的已知界，可得到$\text{KG}(n,k)$的亏格上界为$\binom{n-1}{k}$。2014年，Harvey和Wood证明当$n\neq 4k^2-3k+2$时，$\text{KG}(n,k)$的树宽（亏格的一个下界）为$\binom{n-1}{k}-1$。本文利用另一个亏格下界—— scramble数，改进了该多项式界，证明当$n\neq \frac{3k^2+k+2}{2}$时，$\text{KG}(n,k)$的亏格恰好为$\binom{n-1}{k}$，并猜测利用均匀边scramble可得到更严格的多项式界。随后将论证推广到广义克内泽尔图族，用相同多项式界计算其scramble数与亏格。
英文摘要
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.
Comments23 pages, 8 figures
