Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Bicoloring covers for graphs and hypergraphs   Let the {\it bicoloring cover number $\chi^c(G)$} 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
$\chi^c(G)$ bicolorings. We investigate the relationship between $\chi^c(G)$,
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 $\chi^c(G)$ and $\chi(G)$, respectively. We show that $\chi^c(G)$
can be approximated by a polynomial time algorithm achieving approximation
ratio $\frac{1}{1-t}$, if $\gamma(G)=n^t$, where $t<1$. We also construct a
particular class of hypergraphs $G(V,E)$ called {\it cover friendly}
hypergraphs where the ratio of $\alpha(G)$ 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 $\chi^c(G) > t$. 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.

27
The categories of this paper are: comp-gas
10
The categories of this paper are: q-bio
4
The categories of this paper are: cond-mat
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat
19
The categories of this paper are: eess
26
The categories of this paper are: bayes-an
17
The categories of this paper are: q-fin
29
The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
18
The categories of this paper are: econ
25
The categories of this paper are: patt-sol
6
The categories of this paper are: astro-ph