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Register a new userDomination numbers and noncover complexes of hypergraphs
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Let $mathcal{H}$ be a hypergraph on a finite set $V$. A {em cover} of $mathcal{H}$ is a set of vertices that meets all edges of $mathcal{H}$. If $W$ is not a cover of $mathcal{H}$, then $W$ is said to be a {em noncover} of $mathcal{H}$. The {em noncover complex} of $mathcal{H}$ is the abstract simplicial complex whose faces are the noncovers of $mathcal{H}$. In this paper, we study homological properties of noncover complexes of hypergraphs. In particular, we obtain an upper bound on their Leray numbers. The bound is in terms of hypergraph domination numbers. Also, our proof idea is applied to compute the homotopy type of the noncover complexes of certain uniform hypergraphs, called {em tight paths} and {em tight cycles}. This extends to hypergraphs known results on graphs.
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The power domination problem seeks to find the placement of the minimum number of sensors needed to monitor an electric power network. We generalize the power domination problem to hypergraphs using the infection rule from Bergen et al: given an initial set of observed vertices, $S_0$, a set $Asubseteq S_0$ may infect an edge $e$ if $Asubseteq e$ and for any unobserved vertex $v$, if $Acup {v}$ is contained in an edge, then $vin e$. We combine a domination step with this infection rule to create emph{infectious power domination}. We compare this new parameter to the previous generalization by Chang and Roussel. We provide general bounds and determine the impact of some hypergraph operations.
Let $mathcal{H}$ be a hypergraph of rank $r$. We show that the simplicial complex whose simplices are the hypergraphs $mathcal{F}subsetmathcal{H}$ with covering number at most $p$ is $left(binom{r+p}{r}-1right)$-collapsible, and the simplicial complex whose simplices are the pairwise intersecting hypergraphs $mathcal{F}subsetmathcal{H}$ is $frac{1}{2}binom{2r}{r}$-collapsible.
Let $K$ be a simplicial complex on vertex set $V$. $K$ is called $d$-Leray if the homology groups of any induced subcomplex of $K$ are trivial in dimensions $d$ and higher. $K$ is called $d$-collapsible if it can be reduced to the void complex by sequentially removing a simplex of size at most $d$ that is contained in a unique maximal face. We define the $t$-tolerance complex of $K$, $mathcal{T}_t(K)$, as the simplicial complex on vertex set $V$ whose simplices are formed as the union of a simplex in $K$ and a set of size at most $t$. We prove that for any $d$ and $t$ there exists a positive integer $h(t,d)$ such that, for every $d$-collapsible complex $K$, the $t$-tolerance complex $mathcal{T}_t(K)$ is $h(t,d)$-Leray. The definition of the complex $mathcal{T}_t(K)$ is motivated by results of Montejano and Oliveros on tolerant
The domination number of a graph $G = (V,E)$ is the minimum cardinality of any subset $S subset V$ such that every vertex in $V$ is in $S$ or adjacent to an element of $S$. Finding the domination numbers of $m$ by $n$ grids was an open problem for nearly 30 years and was finally solved in 2011 by Goncalves, Pinlou, Rao, and Thomasse. Many variants of domination number on graphs have been defined and studied, but exact values have not yet been obtained for grids. We will define a family of domination theories parameterized by pairs of positive integers $(t,r)$ where $1 leq r leq t$ which generalize domination and distance domination theories for graphs. We call these domination numbers the $(t,r)$ broadcast domination numbers. We give the exact values of $(t,r)$ broadcast domination numbers for small grids, and we identify upper bounds for the $(t,r)$ broadcast domination numbers for large grids and conjecture that these bounds are tight for sufficiently large grids.
For $ngeq s> rgeq 1$ and $kgeq 2$, write $n rightarrow (s)_{k}^r$ if every hyperedge colouring with $k$ colours of the complete $r$-uniform hypergraph on $n$ vertices has a monochromatic subset of size $s$. Improving upon previous results by textcite{AGLM14} and textcite{EHMR84} we show that [ text{if } r geq 3 text{ and } n rightarrow (s)_k^r text{ then } 2^n rightarrow (s+1)_{k+3}^{r+1}. ] This yields an improvement for some of the known lower bounds on multicolour hypergraph Ramsey numbers. Given a hypergraph $H=(V,E)$, we consider the Ramsey-like problem of colouring all $r$-subsets of $V$ such that no hyperedge of size $geq r+1$ is monochromatic. We provide upper and lower bounds on the number of colours necessary in terms of the chromatic number $chi(H)$. In particular we show that this number is $O(log^{(r-1)} (r chi(H)) + r)$.
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