Coloring the complements of intersection graphs of geometric figures

AbstractLet G¯ be the complement of the intersection graph G of a family of translations of a compact convex figure in Rn. When n=2, we show that χ(G¯)⩽min{3α(G)-2,6γ(G)}, where γ(G) is the size of the minimum dominating set of G. The bound on χ(G¯)⩽6γ(G) is sharp. For higher dimension we show that χ(G¯)⩽⌈2(n2-n+1)1/2⌉n-1⌈(n2-n+1)1/2⌉(α(G)-1)+1, for n⩾3. We also study the chromatic number of the complement of the intersection graph of homothetic copies of a fixed convex body in Rn

Density of Range Capturing Hypergraphs

For a finite set $X$ of points in the plane, a set $S$ in the plane, and a positive integer $k$, we say that a $k$-element subset $Y$ of $X$ is captured by $S$ if there is a homothetic copy $S'$ of $S$ such that $X\cap S' = Y$, i.e., $S'$ contains exactly $k$ elements from $X$. A $k$-uniform $S$-capturing hypergraph $H = H(X,S,k)$ has a vertex set $X$ and a hyperedge set consisting of all $k$-element subsets of $X$ captured by $S$. In case when $k=2$ and $S$ is convex these graphs are planar graphs, known as convex distance function Delaunay graphs. In this paper we prove that for any $k\geq 2$, any $X$, and any convex compact set $S$, the number of hyperedges in $H(X,S,k)$ is at most $(2k-1)|X| - k^2 + 1 - \sum_{i=1}^{k-1}a_i$, where $a_i$ is the number of $i$-element subsets of $X$ that can be separated from the rest of $X$ with a straight line. In particular, this bound is independent of $S$ and indeed the bound is tight for all "round" sets $S$ and point sets $X$ in general position with respect to $S$. This refines a general result of Buzaglo, Pinchasi and Rote stating that every pseudodisc topological hypergraph with vertex set $X$ has $O(k^2|X|)$ hyperedges of size $k$ or less.Comment: new version with a tight result and shorter proo

Transversal numbers of translates of a convex body

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