Toric algebra of hypergraphs

The edges of any hypergraph parametrize a monomial algebra called the edge subring of the hypergraph. We study presentation ideals of these edge subrings, and describe their generators in terms of balanced walks on hypergraphs. Our results generalize those for the defining ideals of edge subrings of graphs, which are well-known in the commutative algebra community, and popular in the algebraic statistics community. One of the motivations for studying toric ideals of hypergraphs comes from algebraic statistics, where generators of the toric ideal give a basis for random walks on fibers of the statistical model specified by the hypergraph. Further, understanding the structure of the generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in algebraic statistics and to combinatorial discrepancy. Section 6 (open problems) has been moderately revise

On the (non-)existence of polynomial kernels for Pl-free edge modification problems

Given a graph G = (V,E) and an integer k, an edge modification problem for a graph property P consists in deciding whether there exists a set of edges F of size at most k such that the graph H = (V,E \vartriangle F) satisfies the property P. In the P edge-completion problem, the set F of edges is constrained to be disjoint from E; in the P edge-deletion problem, F is a subset of E; no constraint is imposed on F in the P edge-edition problem. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size k of the edge set F, it has been proved that if P is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three P edge-modification problems are FPT. It was then natural to ask whether these problems also admit a polynomial size kernel. Using recent lower bound techniques, Kratsch and Wahlstrom answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlstrom asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of P4-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that parameterized cograph edge modification problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the Pl-free and Cl-free edge-deletion problems for large enough l

Large Cross-free sets in Steiner triple systems

A {\em cross-free} set of size $m$ in a Steiner triple system $(V,{\cal{B}})$ is three pairwise disjoint $m$-element subsets $X_1,X_2,X_3\subset V$ such that no $B\in {\cal{B}}$ intersects all the three $X_i$-s. We conjecture that for every admissible $n$ there is an STS$(n)$ with a cross-free set of size $\lfloor{n-3\over 3}\rfloor$ which if true, is best possible. We prove this conjecture for the case $n=18k+3$, constructing an STS$(18k+3)$ containing a cross-free set of size $6k$. We note that some of the $3$-bichromatic STSs, constructed by Colbourn, Dinitz and Rosa, have cross-free sets of size close to $6k$ (but cannot have size exactly $6k$). The constructed STS$(18k+3)$ shows that equality is possible for $n=18k+3$ in the following result: in every $3$-coloring of the blocks of any Steiner triple system STS$(n)$ there is a monochromatic connected component of size at least $\lceil{2n\over 3}\rceil+1$ (we conjecture that equality holds for every admissible $n$). The analogue problem can be asked for $r$-colorings as well, if r-1 \equiv 1,3 \mbox{ (mod 6)} and $r-1$ is a prime power, we show that the answer is the same as in case of complete graphs: in every $r$-coloring of the blocks of any STS$(n)$, there is a monochromatic connected component with at least ${n\over r-1}$ points, and this is sharp for infinitely many $n$.Comment: Journal of Combinatorial Designs, 201

On unimodular hypergraphs

Call number: LD2668 .T4 1985 G63Master of Scienc

On the (non-)existence of polynomial kernels for PlP_l-free edge modification problems

International audienceGiven a graph $G=(V,E)$ and a positive integer $k$, an edge modification problem for a graph property $\Pi$ consists in deciding whether there exists a set $F$ of pairs of $V$ of size at most $k$ such that the graph $H=(V,E\vartriangle F)$ satisfies the property $\Pi$. In the $\Pi$ \emph{edge-completion problem}, the set $F$ is constrained to be disjoint from $E$; in the $\Pi$ \emph{edge-deletion problem}, $F$ is a subset of $E$; no constraint is imposed on $F$ in the $\Pi$ \emph{edge-editing problem}. A number of optimization problems can be expressed in terms of graph modification problems which have been extensively studied in the context of parameterized complexity. When parameterized by the size $k$ of the set $F$, it has been proved that if $\Pi$ is an hereditary property characterized by a finite set of forbidden induced subgraphs, then the three $\Pi$ edge-modification problems are FPT. It was then natural to ask whether these problems also admit a polynomial kernel. Using recent lower bound techniques, Kratsch and Wahlström answered this question negatively. However, the problem remains open on many natural graph classes characterized by forbidden induced subgraphs. Kratsch and Wahlström asked whether the result holds when the forbidden subgraphs are paths or cycles and pointed out that the problem is already open in the case of $P_4$-free graphs (i.e. cographs). This paper provides positive and negative results in that line of research. We prove that \textsc{Parameterized cograph edge-modification} problems have cubic vertex kernels whereas polynomial kernels are unlikely to exist for the \textsc{$P_l$-free edge-deletion} and the \textsc{$C_l$-free edge-deletion} problems for $l\geq 7$ and $l\geq 4$ respectively. Indeed, if they exist, then $NP \subseteq coNP / poly$