Bicoloring covers for graphs and hypergraphs

TL;DR

This paper studies the bicoloring cover number of hypergraphs, establishing bounds, relationships with other parameters, and providing approximation algorithms, including for special classes like cover friendly hypergraphs.

Contribution

Introduces the concept of cover independence number, derives bounds for bicoloring cover number, and develops approximation algorithms with theoretical guarantees.

Findings

Lower bounds for $ ext{chi}^c(G)$ and $ ext{chi}(G)$ based on $ ext{gamma}(G)$.

An approximation algorithm with ratio depending on $ ext{gamma}(G)$.

Construction of cover friendly hypergraphs with large ratios of $ ext{alpha}(G)$ to $ ext{gamma}(G)$.

Abstract

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.