An old result of Tutte states that any $d$-regular graph contains a spanning subgraph in which every vertex has degree $k$ or $k+1$, for every $1\leq k\leq d$. We generalize this statement to hypergraphs, showing, for example, that every $3$-uniform $d$-regular hypergraph contains a subgraph in which all degrees are $k, k+1$ or $k+2$, for every $1\leq k\leq d$. This statement is best possible in the sense that the corresponding statement with only two allowed consecutive values is not true. We provide generalizations of this statement to higher uniformities and discuss several open problems.
A balanced $k$-partite $k$-graph is a $k$-uniform hypergraph whose vertex set is partitioned into $k$ classes of the same size and whose edges meet every class in exactly one vertex. Lo and Markstr\"om (2014) determined the minimum vertex-degree threshold for perfect matchings when $k=3$, and Lu, Wang and Yuan recently...
Jie Han, Hong-Liang Lu, Bin Wang et al.· 0 citations
We prove that the number of labelled embeddings of any $n$-vertex tree $T$ into an $n$-vertex graph $G$ of maximum degree $d$ satisfies $$ \mathrm{inj}(T,G) \leq (d/e)^n \exp(o_d(1) n). $$ The bound is sharp up to determining $o_d(1)$, even for paths, and the dependence of the error $\exp(o_d(1)n)$ on $d$ is necessary....
Csongor Beke, V. Boskovic, Nina Kamčev et al.· 0 citations
For fixed integers $k\ge3$ and $1\le d\le k-1$ and sufficiently large $n\in k\mathbb N$, we establish the sharp minimum $d$-degree thresholds that forces perfect matching in every $n$-vertex $k$-uniform hypergraphs. This was conjectued by Treglown and Zhao, and the $d=1$ case was conjectued by K\"uhn, Osthus and Treglo...
Jie Han, Hong-Liang Lu, Bin Wang et al.· 2 citations
We extend the hypergraph regularity method to sparse $3$-uniform hypergraphs whose vertex links are $C_4$-free. In other words, we consider hypergraphs $H=(V,E)$ that are $K_{1,2,2}$-free, which implies that $|E|=O(|V|^{5/2})$. For such hypergraphs we establish a sparse analogue of the removal lemma for the tight cycle...
Ayush Basu, C. Reiher, V. Rödl et al.· 0 citations
We prove that, for every integer $k\ge 2$, there exists a constant $c_k>0$ such that every graph on $n\ge R(k,k)$ vertices with maximum degree $\Delta$ contains an induced subgraph on at least $n-c_k\sqrt{\Delta}$ vertices whose maximum degree is attained by at least $k$ vertices. This confirms a conjecture of Caro and...
We prove that for every finite connected simple graph $G$ on $n$ vertices with minimum degree $k\geq 2$, there is a coupling of its uniform spanning tree and its random $1$-out subgraph under which the expected number of differing edges is $O(n\log (k)/\sqrt{k})$. More generally, we establish analogous couplings betwee...
András Mészáros, Yuval Peled· 0 citations
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