In a mathematics workshop with $mn$ mathematicians from $n$ different areas, each area consisting of $m$ mathematicians, we want to create a collaboration network. For this purpose, we would like to schedule daily meetings between groups of size three, so that (i) two people of the same area meet one person of another area, (ii) each person has exactly $r$ meeting(s) each day, and (iii) each pair of people of the same area have exactly $λ$ meeting(s) with each person of another area by the end of the workshop. Using hypergraph amalgamation-detachment, we prove a more general theorem. In particular we show that above meetings can be scheduled if: $3 \ | \ rm$, $2 \ | \ rnm$ and $r \ | \ 3λ(n-1)\binom{m}{2}$. This result can be viewed as an analogue of Baranyai's theorem on factorizations of complete multipartite hypergraphs.
For a hypergraph G and a positive integer s, let χ_{\ell} (G,s) be the minimum value of l such that G is L-colorable from every list L with |L(v)|=l for each v\in V(G) and |L(u)\cap L(v)|\leq s for all u, v\in e\in E(G). This parameter was studied by Kratochvíl, Tuza and Voigt for various kinds of graphs. Using randomized constructions we find the asymptotics of χ_{\ell} (G,s) for balanced complete multipartite graphs and for complete k-partite k-uniform hypergraphs.
We classify all complete uniform multipartite hypergraphs with respect to some algebraic properties, such as being (almost) complete intersection, Gorenstein, level, $l$-Cohen-Macaulay, $l$-Buchsbaum, unmixed, and satisfying Serre's condition $S_r$, via some combinatorial terms. Also, we prove that for a complete $s$-uniform $t$-partite hypergraph $H$, vertex decomposability, shellability, sequentially $S_r$ and sequentially Cohen-Macaulay properties coincide with the condition that $H$ has $t-1$ sides consisting of a single vertex. Moreover, we show that the latter condition occurs if and onl…
One of possible interpretations of the well-known König--Hall--Egerváry theorem is a full characterization of all bipartite graphs extremal for fractional matchings of a given weight (or, equivalently, a characterization of $(0,1)$-matrices extremal for partial fractional diagonals of a given length). In this paper we initiate the study of $d$-partite $d$-uniform hypergraphs that are extremal for fractional perfect matchings (or, equivalently, $d$-dimensional $(0,1)$-matrices that are extremal for polydiagonals). For this purpose, we analyze similarities and differences between $2$-dimensional…
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 $A_1,\ldots,A_k$ be a given $k$-partition of $[n]$ with $|A_i|=n_i\geqslant 1$. An $r$-uniform hypergraph $H$ is called {\it $k$-partite} if each edge $e$ satisfies $|e\cap A_i|\leqslant 1$ for $1\leqslant i\leqslant k$. In this paper, the number of linear $k$-partite $r$-uniform hypergraphs on $n\to\infty$ vertices is determined asymptotically when the numbe…
It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the so-called Cartesian skeleton of the graphs under investigation. In this contribution, we show that every connected thin hypergraph H has a unique prime factorization with respect to the normal and strong (hypergraph) product. Both products coincide with the usual strong graph product whenever H is a graph. We introduce the notion of the Cartesian skeleton o…
Let $c$ be an edge-coloring of the complete $n$-vertex graph $K_n$. The problem of finding properly colored and rainbow Hamilton cycles in $c$ was initiated in 1976 by Bollobás and Erd\H os and has been extensively studied since then. Recently it was extended to the hypergraph setting by Dudek, Frieze and Ruciński. We generalize these results, giving sufficient local (resp. global) restrictions on the colorings which guarantee a properly colored (resp. rainbow) copy of a given hypergraph $G$. We also study multipartite analogues of these questions. We give (up to a constant factor) optimal suf…
Consider the following problem. In a school with three classes containing $n$ students each, given that their genders are unknown, find the minimum possible number of triples of same-gender students not all of which are from the same class. Muaengwaeng asked this question and conjectured that the minimum scenario occurs when the classes are all boy, all girl and half-and-half. In this paper, we solve many generalizations of the problem including when the school has more than three classes, when triples are replaced by groups of larger sizes, when the classes are of different sizes, and when ge…
A perfect $K_t$-matching in a graph $G$ is a spanning subgraph consisting of vertex disjoint copies of $K_t$. A classic theorem of Hajnal and Szemerédi states that if $G$ is a graph of order $n$ with minimum degree $δ(G) \ge (t-1)n/t$ and $t| n$, then $G$ contains a perfect $K_t$-matching. Let $G$ be a $t$-partite graph with vertex classes $V_1$,..., $V_t$ each of size $n$. We show that if every vertex $x \in V_i$ is joined to at least $((t-1)/t + γ)n $ vertices of $V_j$ for $i \ne j$, then $G$ contains a perfect $K_t$-matching, thus verifying a conjecture of Fisher asymptotically. Furthermore…
An automorphism group G of a 1-factorization of the complete multipartite graph $K_{m\times n}$ consists in permutations of the vertices of the graph mapping factors to factors. In this paper, we give a complete answer to the existence or non-existence problem of a 1-factorization of $K_{m\times n}$ admitting an abelian group acting sharply transitively on the vertices of the graph.
Let $χ_l(G)$ denote the list chromatic number of the $r$-uniform hypergraph~$G$. Extending a result of Alon for graphs, Saxton and the second author used the method of containers to prove that, if $G$ is simple and $d$-regular, then $χ_l(G)\ge (1/(r-1)+o(1))\log_r d$. To see how close this inequality is to best possible, we examine $χ_l(G)$ when $G$ is a random $r$-partite hypergraph with $n$ vertices in each class. The value when $r=2$ was determined by Alon and Krivelevich, here we show that $χ_l(G)= (g(r,α)+o(1))\log_r d$ almost surely, where $d$ is the expected average degree of~$G$ and $α…
Ryser conjectured that $τ\le(r-1)ν$ for $r$-partite hypergraphs, where $τ$ is the covering number and $ν$ is the matching number. We prove this conjecture for $r\le9$ in the special case of linear intersecting hypergraphs, in other words where every pair of lines meets in exactly one vertex. Aharoni formulated a stronger version of Ryser's conjecture which specified that each $r$-partite hypergraph should have a cover of size $(r-1)ν$ of a particular form. We provide a counterexample to Aharoni's conjecture with $r=13$ and $ν=1$. We also report a number of computational results. For $r=7$, we …
The generalization of the Oberwolfach Problem, proposed by J. Liu in 2000, asks for a uniform $2$-factorization of the complete multipartite graph $K_{m\times n}$. Here we focus our attention on $2$-factorizations regular under the cyclic group $Z_{mn}$, whose $2$-factors are disjoint union of cycles all of even length $\ell$. In particular, we present a complete solution for the extremal cases $\ell=4$ and $\ell=mn$.
A famous conjecture of Ryser is that in an $r$-partite hypergraph the covering number is at most $r-1$ times the matching number. If true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. We show that the conjecture, if true, is also sharp for the smallest previously open value, namely $r=7$. For $r\in\{6,7\}$, we find the minimal number $f(r)$ of edges in an intersecting $r$-partite hypergraph that has covering number at least $r-1$. We find that $f(r)$ is achieved only by linear hypergraphs for $r\le5$, but that this is not the case for $r\in\{6,7\}…
Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an \emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor problems in quasi-random $k$-graphs with minimum degree $Ω(n^{k-1})$. In particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs $H(n)$ with minimum degree $Ω(n^2)$ and minimum codegree $Ω(n)$ but with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite $3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense …
A 1-factor of a hypergraph $G=(X,W)$ is a set of hyperedges such that every vertex of $G$ is incident to exactly one hyperedge from the set. A 1-factorization is a partition of all hyperedges of $G$ into disjoint 1-factors. The adjacency matrix of a $d$-uniform hypergraph $G$ is the $d$-dimensional (0,1)-matrix of order $|X|$ such that an element $a_{α_1, \ldots, α_d}$ of $A$ equals 1 if and only if $\left\{α_1, \ldots, α_d\right\}$ is a hyperedge of $G$. Here we estimate the number of 1-factors of uniform hypergraphs and the number of 1-factorizations of complete uniform hypergraphs by means …
In this paper, two results are obtained on a hypergraph embedding problem. The proof technique is itself of interest, being the first time amalgamations have been used to address the embedding of hypergraphs. The first result finds necessary and sufficient conditions for the embedding a hyperedge-colored copy of the complete 3-uniform hypergraph of order $m$, $K_m^3$, into an $r$-factorization of $K_n^3$, providing that $n> 2m+(-1+\sqrt{8m^2-16m-7})/2$. The second result finds necessary and sufficient conditions for an embedding when not only are the colors of the hyperedges of $K_m^3$ given, …
For $r:=(r_1,\dots,r_k)$, an $r$-factorization of the complete $λ$-fold $h$-uniform $n$-vertex hypergraph $λK_n^h$ is a partition of (the edges of) $λK_n^h$ into $F_1,\dots, F_k$ such that for $i=1,\dots,k$, $F_i$ is $r_i$-regular and spanning. Suppose that $n \geq (h-1)(2m-1)$. Given a partial $r$-factorization of $λK_m^h$, that is, a coloring (i.e. partition) $P$ of the edges of $λK_m^h$ into $F_1,\dots, F_k$ such that for $i=1,\dots,k$, $F_i$ is spanning and the degree of each vertex in $F_i$ is at most $r_i$, we find necessary and sufficient conditions that ensure $P$ can be extended to a …
A detachment of a hypergraph $\scr F$ is a hypergraph obtained from $\scr F$ by splitting some or all of its vertices into more than one vertex. Amalgamating a hypergraph $\scr G$ can be thought of as taking $\scr G$, partitioning its vertices, then for each element of the partition squashing the vertices to form a single vertex in the amalgamated hypergraph $\scr F$. In this paper we use Nash-Williams lemma on laminar families to prove a detachment theorem for amalgamated 3-uniform hypergraphs, which yields a substantial generalization of previous amalgamation theorems by Hilton, Rodger and N…
We generalize the Ihara-Selberg zeta function to hypergraphs in a natural way. Hashimoto's factorization results for biregular bipartite graphs apply, leading to exact factorizations. For $(d,r)$-regular hypergraphs, we show that a modified Riemann hypothesis is true if and only if the hypergraph is Ramanujan in the sense of Winnie Li and Patrick Solé. Finally, we give an example to show how the generalized zeta function can be applied to graphs to distinguish non-isomorphic graphs with the same Ihara-Selberg zeta function.
A \textit{distinguishing partition} of a group $X$ with automorphism group ${aut}(X)$ is a partition of $X$ that is fixed by no nontrivial element of ${aut}(X)$. In the event that $X$ is a complete multipartite graph with its automorphism group, the existence of a distinguishing partition is equivalent to the existence of an asymmetric hypergraph with prescribed edge sizes. An asymptotic result is proven on the existence of a distinguishing partition when $X$ is a complete multipartite graph with $m_1$ parts of size $n_1$ and $m_2$ parts of size $n_2$ for small $n_1$, $m_2$ and large $m_1$, $n…
We show that additive induced-hereditary properties of coloured hypergraphs can be uniquely factorised into irreducible factors. Our constructions and proofs are so general that they can be used for arbitrary concrete categories of combinatorial objects; we provide some examples of such combinatorial objects.
A new, constructive proof with a small explicit constant is given to the Erdős-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most $O(n/\log n)$ times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for graphs and for uniform hypergraphs. We show that these latter constants cannot be improved by more than a factor of 1.89 even for fractional covering by arbitrary complete multipartite subgraphs or…
El-Zanati et al proved that for any 1-factorization $\mathcal{F}$ of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor. We generalize their result and show that in any proper coloring of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor.
