Related papers: Bounds on the Game Transversal Number in Hypergrap…

The Tur\'{a}n number of a graph $H$, $\text{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph that does not contain $H$ as a subgraph. For a vertex $v$ and a multi-set $\mathcal{F}$ of graphs, the suspension $\mathcal{F}+v$…

We consider a cops and robber game where the cops are blocking edges of a graph, while the robber occupies its vertices. At each round of the game, the cops choose some set of edges to block and right after the robber is obliged to move to…

In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph $T$ consisting of three edges $e,f$ and $g$ such that $|e \cap f| = |f…

Let $H=(V,E)$ be a hypergraph with vertex set $V$ and edge set $E$. $S\subseteq V$ is a feedback vertex set (FVS) of $H$ if $H\setminus S$ has no cycle and $\tau_c(H)$ denote the minimum cardinality of a FVS of $H$. In this paper, we prove…

For an integer $r\geqslant 3$, a hypergraph on vertex set $[n]$ is $r$-uniform if each edge is a set of $r$ vertices, and is said to be linear if every two distinct edges share at most one vertex. Given a family $\mathcal{H}$ of linear…

In a Take-Away Game on hypergraphs, two players take turns to remove the vertices and the hyperedges of the hypergraphs. In each turn, a player must remove either a single vertex or a hyperedge. When a player chooses to remove one vertex,…

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 transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponential-time algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed k>2, our algorithms for…

Eternal vertex cover is the following two-player game between a defender and an attacker on a graph. Initially, the defender positions k guards on k vertices of the graph; the game then proceeds in turns between the defender and the…

A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway's conjecture states that a thrackle has at most as many edges as vertices. In this…

The independence number $\alpha(H)$ of a hypergraph $H$ is the maximum cardinality of a set of vertices of $H$ that does not contain an edge of $H$. Generalizing Shearer's classical lower bound on the independence number of triangle-free…

We say that a hypergraph $\mathcal{H}$ contains a graph $H$ as a trace if there exists some set $S\subset V(\mathcal{H})$ such that $\mathcal{H}|_S=\{h\cap S: h\in E(\mathcal{H})\}$ contains a subhypergraph isomorphic to $H$. We study the…

A $k$-uniform hypergraph $H = (V, E)$ is called $\ell$-orientable, if there is an assignment of each edge $e\in E$ to one of its vertices $v\in e$ such that no vertex is assigned more than $\ell$ edges. Let $H_{n,m,k}$ be a hypergraph,…

Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then hypergraph H is invertible iff there exists a permutation pi of V such that for all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility critical if H…

Let $r \ge 2$ be a fixed constant and let $ {\mathcal H}$ be an $r$-uniform, $D$-regular hypergraph on $N$ vertices. Assume further that $ D \to \infty$ as $N \to \infty$ and that degrees of pairs of vertices in ${\mathcal H}$ are at most…

The distinguishing number of a graph $G$ is a symmetry related graph invariant whose study started two decades ago. The distinguishing number $D(G)$ is the least integer $d$ such that $G$ has a $d$-distinguishing coloring. A distinguishing…

We consider the following definition of connectivity in $k$-uniform hypergraphs: Two $j$-sets are $j$-connected if there is a walk of edges between them such that two consecutive edges intersect in at least $j$ vertices. We determine the…

Game coloring is a well-studied two-player game in which each player properly colors one vertex of a graph at a time until all the vertices are colored. An `eternal' version of game coloring is introduced in this paper in which the vertices…

In this paper, we study Maker-Breaker games on the random hypergraph $H_{n,s,p}$, obtained from the complete $s$-graph by keeping every edge independently with probability $p$. We determine the threshold probability for the property of…

The domination game is played on a graph $G$ by two players, named Dominator and Staller. They alternatively select vertices of $G$ such that each chosen vertex enlarges the set of vertices dominated before the move on it. Dominator's goal…