arXiv:2611.17183 · v1

New bounds on the boxicity of Kneser graphs

Kneser 图 boxicity 的新界

首次提交 2026-11-13 01:51

版本更新 2026-11-13 01:51

雷达观测 2026-11-13 14:15

摘要

For a graph $G=(V,E)$, the boxicity $\operatorname{box}(G)$ is the least integer $d$ such that $G$ is the intersection graph of axis-parallel boxes in $\mathbb{R}^d$. The Kneser graph $K(n,k)$ is the graph whose vertex set corresponds to the family of $k$ sized subsets of an $n$ sized ground set, and a pair of vertices are adjacent if and only if the corresponding sets are disjoint. In this short paper, we provide new bounds for the boxicity of $K(n,k)$. We first show that for $k\ge 2$ and $n\ge 2k+1$, $\operatorname{box}(K(n,k))\ge n-2k$, thereby improving the lower bounds of Caoduro and Lichev [Discrete Mathematics 346(5):113333 (2023)] when $n\geq 3k+1$ and $k\ge 3$. The proof of this result uses the topological method and follows a strategy similar to Greene's proof of the Kneser's conjecture on the chromatic number of $K(n,k)$. We also show that for $k\ge 2$, $t\ge 1$ and $n=2k+t$, $\operatorname{box}(K(2k+t,k)) \leq O( t^3\log(2k+t) ).$ In particular, for every $t=t(k)\ge1$ with $t^3\log k=o(k)$, $\operatorname{box}(K(2k+t,k))=o(k)$. This is the first upper bound on the boxicity of $K(n,k)$ that is sublinear in $n$. We also show that for large enough $k$, $\operatorname{box}(K(2k+1,k))\le \frac{3\ln (2k+1)}{\ln\ln (2k+1)}$.