Spectrum of mixed bi-uniform hypergraphs

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 $D$-edges are not monochromatic. The feasible set $S(H)$ of $H$ is the set of all integers, $s$, such that $H$ has a proper coloring with $s$ colors. Bujt\'as and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a characterization of feasible sets for mixed hypergraphs with all $C$- and $D$-edges of the same size $r$, $r\geq 3$. In this note, we give a short proof of a complete characterization of all possible feasible sets for mixed hypergraphs with all $C$-edges of size $\ell$ and all $D$-edges of size $m$, where $\ell, m \geq 2$. Moreover, we show that for every sequence $(r(s))_{s=\ell}^n$, $n \geq \ell$, of natural numbers there exists such a hypergraph with exactly $r(s)$ proper colorings using $s$ colors, $s = \ell,\ldots,n$, and no proper coloring with more than $n$ colors. Choosing $\ell = m=r$ this answers a question of Bujt\'as and Tuza, and generalizes their result with a shorter proof.Comment: 9 pages, 5 figure

Covering graphs by monochromatic trees and Helly-type results for hypergraphs

How many monochromatic paths, cycles or general trees does one need to cover all vertices of a given $r$-edge-coloured graph $G$? These problems were introduced in the 1960s and were intensively studied by various researchers over the last 50 years. In this paper, we establish a connection between this problem and the following natural Helly-type question in hypergraphs. Roughly speaking, this question asks for the maximum number of vertices needed to cover all the edges of a hypergraph $H$ if it is known that any collection of a few edges of $H$ has a small cover. We obtain quite accurate bounds for the hypergraph problem and use them to give some unexpected answers to several questions about covering graphs by monochromatic trees raised and studied by Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao, Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi

Hypergraphs and hypermatrices with symmetric spectrum

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric $r$-matrix $A$ has a symmetric spectrum if and only if $r$ is even and $A$ is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs with symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu. Separately, similar results are obtained for the $H$-spectram of hypermatrices.Comment: 17 pages. Corrected proof on p. 1

Linear trees in uniform hypergraphs

Given a tree T on v vertices and an integer k exceeding one. One can define the k-expansion T^k as a k-uniform linear hypergraph by enlarging each edge with a new, distinct set of (k-2) vertices. Then T^k has v+ (v-1)(k-2) vertices. The aim of this paper is to show that using the delta-system method one can easily determine asymptotically the size of the largest T^k-free n-vertex hypergraph, i.e., the Turan number of T^k.Comment: Slightly revised, 14 pages, originally presented on Eurocomb 201

Sharp Concentration of Hitting Size for Random Set Systems

Consider the random set system of {1,2,...,n}, where each subset in the power set is chosen independently with probability p. A set H is said to be a hitting set if it intersects each chosen set. The second moment method is used to exhibit the sharp concentration of the minimal size of H for a variety of values of p.Comment: 11 page