Bicoloring covers for graphs and hypergraphs

Abstract: Let the {\it bicoloring cover number $\chi<sup>c(G)$} 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 $\chi<sup>c(G)$ bicolorings. We investigate the relationship between $\chi<sup>c(G)$, matchings, hitting sets, $\alpha(G)$(independence number) and $\chi(G)$ (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 $\gamma(G)$" and prove that $\log \frac{|V|}{\gamma(G)}$ and $\frac{|V|}{2\gamma(G)}$ are lower bounds for $\chi<sup>c(G)$ and $\chi(G)$, respectively. We show that $\chi<sup>c(G)$ can be approximated by a polynomial time algorithm achieving approximation ratio $\frac{1}{1-t}$, if $\gamma(G)=n<sup>t$, where $t&lt;1$. We also construct a particular class of hypergraphs $G(V,E)$ called {\it cover friendly} hypergraphs where the ratio of $\alpha(G)$ to $\gamma(G)$ 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} $\omega(G)=k$ and $\chi^c(G) &gt; t$. 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} &lt; m(k,x) \leq x \cdot k² \cdot 2<sup>{(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 $d(G) \leq(\frac{2<sup>{x(k-1)}}{e}-1)$ using $nx+kx\frac{m}{d}$ random bits.