Related papers: Rectilinear Crossings in Complete Balanced d-Parti…

For all integers $k,d$ such that $k \geq 3$ and $k/2\leq d \leq k-1$, let $n$ be a sufficiently large integer {\rm(}which may not be divisible by $k${\rm)} and let $s\le \lfloor n/k\rfloor-1$. We show that if $H$ is a $k$-uniform hypergraph…

The classical Ham Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single affine hyperplane. A generalization of Dolnikov asserts that any $d$ families of pairwise intersecting compact,…

A d-partite hypergraph is called *fractionally balanced* if there exists a non-negative, not identically zero, function on its edge set that has constant degrees in each vertex side. Using a topological version of Hall's theorem we prove…

An $n$-vertex $k$-uniform hypergraph $G$ is $(d,\alpha)$-degenerate if $m_1(G)\le{d}$ and there exists a constant $\varepsilon >0$ such that for every subset $U\subseteq{V(G)}$ with size $2\le|U|\le{\varepsilon n}$, we have…

A graph $\Gamma$ is said to be universal for a class of graphs $\mathcal{H}$ if $\Gamma$ contains a copy of every $H \in \mathcal{H}$ as a subgraph. The number of edges required for a host graph $\Gamma$ to be universal for the class of…

Let $k\ge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $\binom{n-1}{k-1} + \lfloor\frac{n-1}{k}…

A digraph is {\em $d$-dominating} if every set of at most $d$ vertices has a common out-neighbor. For all integers $d\geq 2$, let $f(d)$ be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete…

A mixed hypergraph is a triple $H=(V,\mathcal{C},\mathcal{D})$, where $V$ is a set of vertices, $\mathcal{C}$ and $\mathcal{D}$ are sets of hyperedges. A vertex-coloring of $H$ is proper if $C$-edges are not totally multicolored and…

For any natural number $d$, a graph $G$ is a (disjoint) $d$-interval graph if it is the intersection graph of (disjoint) $d$-intervals, the union of $d$ (disjoint) intervals on the real line. Two important subclasses of $d$-interval graphs…

A conjecture due to the fourth author states that every $d$-regular planar multigraph can be $d$-edge-coloured, provided that for every odd set $X$ of vertices, there are at least $d$ edges between $X$ and its complement. For $d = 3$ this…

The contact graph of a packing of translates of a convex body in Euclidean $d$-space $\mathbb E^d$ is the simple graph whose vertices are the members of the packing, and whose two vertices are connected by an edge if the two members touch…

We show that, for any graph $F$ and $\eta>0$, there exists a $d_0=d_0(F,\eta)$ such that every $n$-vertex $d$-regular graph with $d \geq d_0$ has a collection of vertex-disjoint $F$-subdivisions covering at least $(1-\eta)n$ vertices. This…

A well-known special case of a conjecture attributed to Ryser states that k-partite intersecting hypergraphs have transversals of at most k-1 vertices. An equivalent form was formulated by Gy\'arf\'as: if the edges of a complete graph K are…

We give a new proof of K\"onig's theorem and generalize the Gallai-Edmonds decomposition to balanced hypergraphs in two different ways. Based on our decompositions we give two new characterizations of balanced hypergraphs and show some…

A $k$-uniform hypergraph $H = (V, E)$ is $k$-partite if $V$ can be partitioned into $k$ sets $V_1, \ldots, V_k$ such that every edge in $E$ contains precisely one vertex from each $V_i$. We call such a graph $n$-balanced if $|V_i| = n$ for…

Inspired by earlier results about proper and polychromatic coloring of hypergraphs, we investigate such colorings of directed hypergraphs, that is, hypergraphs in which the vertices of each hyperedge is partitioned into two parts, a tail…

In this paper, we characterize all unmixed d-uniform r-partite hypergraphs under a certain condition. Also we give a necessary condition for unmixedness in d-uniform hypergraphs with a perfect matching of size n. Finally we give a…

A hypergraph is \textit{bipartite with bipartition $(A, B)$} if every edge has exactly one vertex in $A$, and a matching in such a hypergraph is \textit{$A$-perfect} if it saturates every vertex in $A$. We prove an upper bound on the number…

We give a complete combinatorial characterization of weakly $d$-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in $\mathbb…

Let $n \equiv 0\, (\, \text{mod } 3\,)$ and $H_{n, n/3}^2$ be the 3-graph of order $n$, whose vertex set is partitioned into two sets $S$ and $T$ of size $\frac{1}{3}n+1$ and $\frac{2}{3}n -1$, respectively, and whose edge set consists of…