An asymptotic bound for the strong chromatic number

The strong chromatic number $\chi_{\text{s}}(G)$ of a graph $G$ on $n$ vertices is the least number $r$ with the following property: after adding $r \lceil n/r \rceil - n$ isolated vertices to $G$ and taking the union with any collection of spanning disjoint copies of $K_r$ in the same vertex set, the resulting graph has a proper vertex-colouring with $r$ colours. We show that for every $c > 0$ and every graph $G$ on $n$ vertices with $\Delta(G) \ge cn$, $\chi_{\text{s}}(G) \leq (2 + o(1)) \Delta(G)$, which is asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu

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

Decomposing 1-Sperner hypergraphs

A hypergraph is Sperner if no hyperedge contains another one. A Sperner hypergraph is equilizable (resp., threshold) if the characteristic vectors of its hyperedges are the (minimal) binary solutions to a linear equation (resp., inequality) with positive coefficients. These combinatorial notions have many applications and are motivated by the theory of Boolean functions and integer programming. We introduce in this paper the class of $1$-Sperner hypergraphs, defined by the property that for every two hyperedges the smallest of their two set differences is of size one. We characterize this class of Sperner hypergraphs by a decomposition theorem and derive several consequences from it. In particular, we obtain bounds on the size of $1$-Sperner hypergraphs and their transversal hypergraphs, show that the characteristic vectors of the hyperedges are linearly independent over the reals, and prove that $1$-Sperner hypergraphs are both threshold and equilizable. The study of $1$-Sperner hypergraphs is motivated also by their applications in graph theory, which we present in a companion paper

Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic

The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs $\mathcal{G}$ and $\mathcal{H}$, decide whether $\mathcal{H}$ consists precisely of all minimal transversals of $\mathcal{G}$ (in which case we say that $\mathcal{G}$ is the dual of $\mathcal{H}$). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in $\mathrm{GC}(\log^2 n,\mathrm{PTIME})$, where $\mathrm{GC}(f(n),\mathcal{C})$ denotes the complexity class of all problems that after a nondeterministic guess of $O(f(n))$ bits can be decided (checked) within complexity class $\mathcal{C}$. It was conjectured that non-DUAL is in $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class $\mathrm{GC}(\log^2 n,\mathrm{TC}^0)$ which is a subclass of $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. We here refer to the logtime-uniform version of $\mathrm{TC}^0$, which corresponds to $\mathrm{FO(COUNT)}$, i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for non-DUAL that requires to guess $O(\log^2 n)$ bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in $\mathrm{FO(COUNT)}$. From this result, by the well known inclusion $\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}$, it follows that DUAL belongs also to $\mathrm{DSPACE}[\log^2 n]$. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs $\mathcal{G}$ and $\mathcal{H}$, computes in quadratic logspace a transversal of $\mathcal{G}$ missing in $\mathcal{H}$.Comment: Restructured the presentation in order to be the extended version of a paper that will shortly appear in SIAM Journal on Computin

A polynomial regularity lemma for semi-algebraic hypergraphs and its applications in geometry and property testing

Fox, Gromov, Lafforgue, Naor, and Pach proved a regularity lemma for semi-algebraic $k$-uniform hypergraphs of bounded complexity, showing that for each $\epsilon>0$ the vertex set can be equitably partitioned into a bounded number of parts (in terms of $\epsilon$ and the complexity) so that all but an $\epsilon$-fraction of the $k$-tuples of parts are homogeneous. We prove that the number of parts can be taken to be polynomial in $1/\epsilon$. Our improved regularity lemma can be applied to geometric problems and to the following general question on property testing: is it possible to decide, with query complexity polynomial in the reciprocal of the approximation parameter, whether a hypergraph has a given hereditary property? We give an affirmative answer for testing typical hereditary properties for semi-algebraic hypergraphs of bounded complexity

Antichain cutsets of strongly connected posets

Rival and Zaguia showed that the antichain cutsets of a finite Boolean lattice are exactly the level sets. We show that a similar characterization of antichain cutsets holds for any strongly connected poset of locally finite height. As a corollary, we get such a characterization for semimodular lattices, supersolvable lattices, Bruhat orders, locally shellable lattices, and many more. We also consider a generalization to strongly connected hypergraphs having finite edges.Comment: 12 pages; v2 contains minor fixes for publicatio

Covering complete partite hypergraphs by monochromatic components

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 colored with k colors then the vertex set of K can be covered by at most k-1 sets, each connected in some color. It turned out that the analogue of the conjecture for hypergraphs can be answered: Z. Kir\'aly proved that in every k-coloring of the edges of the r-uniform complete hypergraph K^r (r >= 3), the vertex set of K^r can be covered by at most $\lceil k/r \rceil$ sets, each connected in some color. Here we investigate the analogue problem for complete r-uniform r-partite hypergraphs. An edge coloring of a hypergraph is called spanning if every vertex is incident to edges of any color used in the coloring. We propose the following analogue of Ryser conjecture. In every spanning (r+t)-coloring of the edges of a complete r-uniform r-partite hypergraph, the vertex set can be covered by at most t+1 sets, each connected in some color. Our main result is that the conjecture is true for 1 <= t <= r-1. We also prove a slightly weaker result for t >= r, namely that t+2 sets, each connected in some color, are enough to cover the vertex set. To build a bridge between complete r-uniform and complete r-uniform r-partite hypergraphs, we introduce a new notion. A hypergraph is complete r-uniform (r,l)-partite if it has all r-sets that intersect each partite class in at most l vertices. Extending our results achieved for l=1, we prove that for any r >= 3, 2 <= l = 1+r-l, in every spanning k-coloring of the edges of a complete r-uniform (r,l)-partite hypergraph, the vertex set can be covered by at most 1+\lfloor \frac{k-r+\ell-1}{\ell}\rfloor sets, each connected in some color.Comment: 14 page