Related papers: A Harary-Sachs Theorem for Hypergraphs

In this paper we consider the problem to reconstruct a $k$-uniform hypergraph from its line graph. In general this problem is hard. We solve this problem when the number of hyperedges containing any pair of vertices is bounded. Given an…

We study off-diagonal Ramsey numbers $r(H, K_n^{(k)})$ of $k$-uniform hypergraphs, where $H$ is a fixed linear $k$-uniform hypergraph and $K_n^{(k)}$ is complete on $n$ vertices. Recently, Conlon et al.\ disproved the folklore conjecture…

The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0 depends…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

We prove an asymptotic formula for the number of $k$-uniform hypergraphs with a given degree sequence, for a wide range of parameters. In particular, we find a formula that is asymptotically equal to the number of $d$-regular $k$-uniform…

For a hypergraph ${\cal H}$, let $P({\cal H},k)$ and $P_l({\cal H},k)$ be its chromatic polynomial and list-color function respectively, and let $\tau'({\cal H})$ be the least non-negative integer $q$ such that $P({\cal H},k)=P_l({\cal…

We give an upper bound on the list chromatic number of a 2-colorable hypergraph which generalizes the bound of Schauz on $k$-partite $k$-uniform hypergraphs. It makes sense for sparse hypergraphs: in particular we show that a $k$-uniform…

In hypergraph theory, determining a characterization of the degree sequence $d=(d_1,d_2,\ldots,d_n)$ where $d_1\ge d_2\ge\ldots,d_n$ are positive integers, of an $h$-uniform simple hypergraph $\cal H$, and deciding the complexity status of…

A famous result by R\"odl, Ruci\'nski, and Szemer\'edi guarantees a (tight) Hamilton cycle in $k$-uniform hypergraphs $H$ on $n$ vertices with minimum $(k-1)$-degree $\delta_{k-1}(H)\geq (1/2+o(1))n$, thereby extending Dirac's result from…

Given a graph $G$, the $k$-coloring graph $\mathcal{C}_k(G)$ is constructed by selecting proper $k$-colorings of $G$ as vertices, with an edge between two colorings if they differ in the color of exactly one vertex. The number of vertices…

A result due to Gy\'arf\'as, Hubenko, and Solymosi (answering a question of Erd\"os) states that if a graph $G$ on $n$ vertices does not contain $K_{2,2}$ as an induced subgraph yet has at least $c\binom{n}{2}$ edges, then $G$ has a…

A $k$-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots,k$. A function from the vertex set of one $k$-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two…

P. Erd\H{o}s [On extremal problems of graphs and generalized graphs, Israel Journal of Mathematics 2 (1964), 183-190] characterised those hypergraphs $F$ that have to appear in any sufficiently large hypergraph $H$ of positive density. We…

We present a simple mechanism, which can be randomised, for constructing sparse $3$-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over $\mathbb{Z}_2^t$ and have vertex degree…

In this paper, we consider the embedding of a complete $d$-uniform geometric hypergraph with $n$ vertices in general position in $\mathbb{R}^d$, where each hyperedge is represented as a $(d-1)$-simplex, and a pair of hyperedges is defined…

A k-uniform hypergraph is algebraic if its vertex set is n-dimensional Euclidean space, for some n, and its hyperedge set is defined from the zero set of some polynomial. The chromatic numbers of all algebraic hypergraphs are determined,…

For any given integer $r\geqslant 3$, let $k=k(n)$ be an integer with $r\leqslant k\leqslant n$. A hypergraph is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if two edges intersect in at most one vertex. Let…

In a seminal paper, Alon and Tarsi have introduced an algebraic technique for proving upper bounds on the choice number of graphs (and thus, in particular, upper bounds on their chromatic number). The upper bound on the choice number of $G$…

Let $k,r \geq 2$ be two integers. We consider the problem of partitioning the hyperedge set of an $r$-uniform hypergraph $H$ into the minimum number $\chi_k'(H)$ of edge-disjoint subhypergraphs in which every vertex has either degree $0$ or…

The chromatic polynomial $\pi_{G}(k)$ of a graph $G$ can be viewed as counting the number of vertices in a family of coloring graphs $\mathcal C_k(G)$ associated with (proper) $k$-colorings of $G$ as a function of the number of colors $k$.…