Bicoloring covers for graphs and hypergraphs

Abstract

Let the {\it bicoloring cover number $χ^c(G)$} for a hypergraph $G(V,E)$ be the minimum number of bicolorings of vertices of $G$ such that every hyperedge $e\in E$ of $G$ is properly bicolored in at least one of the $χ^c(G)$ bicolorings. We investigate the relationship between $χ^c(G)$, matchings, hitting sets, $α(G)$(independence number) and $χ(G)$ (chromatic number). We design a factor $O(\frac{\log n}{\log \log n-\log \log \log n})$ approximation algorithm for computing a bicoloring cover. We define a new parameter for hypergraphs - "cover independence number $γ(G)$" and prove that $\log \frac{|V|}{γ(G)}$ and $\frac{|V|}{2γ(G)}$ are lower bounds for $χ^c(G)$ and $χ(G)$, respectively. We show that $χ^c(G)$ can be approximated by a polynomial time algorithm achieving approximation ratio $\frac{1}{1-t}$, if $γ(G)=n^t$, where $t<1$. We also construct a particular class of hypergraphs $G(V,E)$ called {\it cover friendly} hypergraphs where the ratio of $α(G)$ to $γ(G)$ can be arbitrarily large.We prove that for any $t\geq 1$, there exists a $k$-uniform hypergraph $G$ such that the {\it clique number} $ω(G)=k$ and $χ^c(G) > t$. Let $m(k,x)$ denote the minimum number of hyperedges %in a $k$-uniform hypergraph $G$ such that some $k$-uniform hypergraph $G$ with $m(k,x)$ hyperedges does not have a bicoloring cover of size $x$. We show that $ 2^{(k-1)x-1} < m(k,x) \leq x \cdot k^2 \cdot 2^{(k+1)x+2}$. Let the {\it dependency $d(G)$} of $G$ be the maximum number of hyperedge neighbors of any hyperedge in $G$. We propose an algorithm for computing a bicoloring cover of size $x$ for $G$ if $d(G) \leq(\frac{2^{x(k-1)}}{e}-1)$ using $nx+kx\frac{m}{d}$ random bits.