A Fixed-Parameter Algorithm for the Schrijver Problem

The Schrijver graph $S(n,k)$ is defined for integers $n$ and $k$ with $n \geq 2k$ as the graph whose vertices are all the $k$-subsets of $\{1,2,\ldots,n\}$ that do not include two consecutive elements modulo $n$, where two such sets are adjacent if they are disjoint. A result of Schrijver asserts that the chromatic number of $S(n,k)$ is $n-2k+2$ (Nieuw Arch. Wiskd., 1978). In the computational Schrijver problem, we are given an access to a coloring of the vertices of $S(n,k)$ with $n-2k+1$ colors, and the goal is to find a monochromatic edge. The Schrijver problem is known to be complete in the complexity class $\mathsf{PPA}$. We prove that it can be solved by a randomized algorithm with running time $n^{O(1)} \cdot k^{O(k)}$, hence it is fixed-parameter tractable with respect to the parameter $k$.Comment: 19 pages. arXiv admin note: substantial text overlap with arXiv:2204.0676

On the maximum degree of induced subgraphs of the Kneser graph

For integers $n \geq k \geq 1$, the {\em Kneser graph} $K(n, k)$ is the graph with vertex-set consisting of all the $k$-element subsets of $\{1,2,\ldots,n\}$, where two $k$-element sets are adjacent in $K(n,k)$ if they are disjoint. We show that if $(n,k,s) \in \mathbb{N}^3$ with $n > 10000 k s^5$ and $\mathcal{F}$ is set of vertices of $K(n,k)$ of size larger than $\{A \subset \{1,2,\ldots,n\}:\ |A|=k,\ A \cap \{1,2,\ldots,s\} \neq \varnothing\}$, then the subgraph of $K(n,k)$ induced by $\mathcal{F}$ has maximum degree at least $$\left(1 - O\left(\sqrt{s^3 k/n}\right)\right)\frac{s}{s+1} \cdot {n-k \choose k} \cdot \frac{|\mathcal{F}|}{\binom{n}{k}}.$$ This is sharp up to the behaviour of the error term $O(\sqrt{s^3 k/n})$. In particular, if the triple of integers $(n, k, s)$ satisfies the condition above, then the minimum maximum degree does not increase `continuously' with $|\mathcal{F}|$. Instead, it has $s$ jumps, one at each time when $|\mathcal{F}|$ becomes just larger than the union of $i$ stars, for $i = 1, 2, \ldots, s$. An appealing special case of the above result is that if $\mathcal{F}$ is a family of $k$-element subsets of $\{1,2,\ldots,n\}$ with $|\mathcal{F}| = {n-1 \choose k-1}+1$, then there exists $A \in \mathcal{F}$ such that $\mathcal{F}$ is disjoint from at least $$\left(1/2-O\left(\sqrt{k/n}\right)\right){n-k-1 \choose k-1}$$ of the other sets in $\mathcal{F}$; this is asymptotically sharp if $k=o(n)$. Frankl and Kupavskii, using different methods, have recently proven closely related results under the hypothesis that $n$ is at least quadratic in $k$.Comment: 29 pages. Minor corrections; references added; added an explanation of how the main result essentially solves a problem of Gerbner, Lemons, Palmer, Patk\'os and Sz\'ecs