arXiv subjects
Publications and source records attributed to Bjarne Schülke.
We show that every $k$-uniform hypergraph on $n$ vertices whose minimum $(k-2)$-degree is at least $(5/9+o(1))n^2/2$ contains a Hamiltonian cycle. A construction due to Han and Zhao shows that this minimum degree condition is optimal. The same result was proved independently by Lang and Sahueza-Matamala.
We investigate the problem of determining how many monochromatic trees are necessary to cover the vertices of an edge-coloured random graph. More precisely, we show that for $p\gg n^{-1/6}{(\ln n)}^{1/6}$, in any $3$-edge-colouring of the random graph $G(n,p)$ we can find three monochromatic trees such that their union covers all vertices. This improves, for three colours, a result of Bucić, Korándi and Sudakov.
We show that every 4-uniform hypergraph with $n$ vertices and minimum pair degree at least $(5/9+o(1))n^2/2$ contains a tight Hamiltonian cycle. This degree condition is asymptotically optimal.
Sidorenko's conjecture states that the number of copies of a bipartite graph $H$ in a graph $G$ is asymptotically minimised when $G$ is a quasirandom graph. A notorious example where this conjecture remains open is when $H=K_{5,5}\setminus C_{10}$. It was even unknown whether this graph possesses the strictly stronger, weakly norming property. We take a step towards understanding the graph $K_{5,5}\setminus C_{10}$ by proving that it is not weakly norming. More generally, we show that 'twisted' blow-ups of cycles, which include $K_{5,5}\setminus C_{10}$ and $C_6\square K_2$, are not weakly norming. This answers two questions of Hatami. The method relies on the analysis of Hessian matrices defined by graph homomorphisms, by using the equivalence between the (weakly) norming property and convexity of graph homomorphism densities. We also prove that $K_{t,t}$ minus a perfect matching, proven to be weakly norming by Lovász, is not norming for every $t>3$.
For integers $n\geq k\geq 1$, the Kneser graph $K(n,k)$ is the graph with vertex set $V=[n]^{(k)}$ and edge set $E=\{\{x,y\} \in V^{(2)}: x\cap y=\emptyset\}$. Chen proved that for $n\geq 3k$, Kneser graphs are Hamiltonian and later improved this to $n\geq 2.62k+1$. Furthermore, Chen and Füredi gave a short proof that if $k | n$, Kneser graphs are Hamiltonian for $n\geq 3k$. In this note, we present a short proof that does not need the divisibility condition, i.e., we give a short proof that $K(n,k)$ is Hamiltonian for $n\geq 4k$.
For $k,n\in \mathbb{N}$, the Kneser graph $K(n,k)$ is the graph with vertex set $V=[n]^{(k)}$ and edge set $E=\{\{x,y\} \in V^{(2)}: x\cap y=\emptyset\}$. Chen proved that for $n\geq 3k$, Kneser graphs are Hamiltonian. Similarly as for graphs with Hajnal's and Szemerédi's result about a minimum degree condition for clique factors and the Pósa-Seymour Conjecture together with its solution for large graphs due to Komlós, Sárközy, and Szemerédi, the next step is to ask for clique factors and powers of Hamiltonian cycles in Kneser graphs. For $k,\ell\in \mathbb{N}$, let $n(k,\ell)$ be the smallest integer such that for $n\geq n(k,\ell)$, $K(n,k)$ contains the $\ell$-th power of a Hamiltonian cycle. Katona conjectured that for all but finitely many exceptions, $n(k,\ell)=(\ell+1)k+1$ holds. In particular, it would be interesting to know whether $n(k,\ell)$ is linear in $k$ (for fixed $\ell$). So far this is not known for $k\geq 2$. In this note, we take a first step towards such a linear bound by proving that for $\ell\in \mathbb{N}$, $k\geq \ell$ and $n\geq \ell ^3k$, all but at most $\ell-1$ vertices of $K(n,k)$ can be partitioned into cliques of size $\ell$. Further, we use our methods to extend a short proof due to Chen and Füredi that $K(n,k)$ is Hamiltonian for $n\geq 3k$ and $k \mid n$ to all $n\geq 4k$ if $k\geq 4$.
We consider sufficient conditions for the existence of $k$-th powers of Hamiltonian cycles in $n$-vertex graphs $G$ with minimum degree $μn$ for arbitrarily small $μ>0$. About 20 years ago Komlós, Sarközy, and Szemerédi resolved the conjectures of Pósa and Seymour and obtained optimal minimum degree conditions for this problem by showing that $μ=\frac{k}{k+1}$ suffices for large $n$. For smaller values of $μ$ the given graph $G$ must satisfy additional assumptions. We show that inducing subgraphs of density $d>0$ on linear subsets of vertices and being inseparable, in the sense that every cut has density at least $μ>0$, are sufficient assumptions for this problem and, in fact, for a variant of the bandwidth theorem. This generalises recent results of Staden and Treglown.