Strong $(r,p)$ Cover for Hypergraphs

Abstract: We introduce the notion of the { \it strong $(r,p)$ cover} number $\chi<sup>c(G,k,r,p)$ for $k$-uniform hypergraphs $G(V,E)$, where $\chi<sup>c(G,k,r,p)$ denotes the minimum number of $r$-colorings of vertices in $V$ such that each hyperedge in $E$ contains at least $min(p,k)$ vertices of distinct colors in at least one of the $\chi<sup>c(G,k,r,p)$ $r$-colorings. We derive the exact values of $\chi<sup>c(K_n<sup>k,k,r,p)$ for small values of $n$, $k$, $r$ and $p$, where $K_n<sup>k$ denotes the complete $k$-uniform hypergraph of $n$ vertices. We study the variation of $\chi<sup>c(G,k,r,p)$ with respect to changes in $k$, $r$, $p$ and $n$; we show that $\chi<sup>c(G,k,r,p)$ is at least (i) $\chi<sup>c(G,k,r-1,p-1)$, and, (ii) $\chi<sup>c(G&#39;,k-1,r,p-1)$, where $G&#39;$ is any $(n-1)$-vertex induced sub-hypergraph of $G$. We establish a general upper bound for $\chi<sup>c(K_n<sup>k,k,r,p)$ for complete $k$-uniform hypergraphs using a divide-and-conquer strategy for arbitrary values of $k$, $r$ and $p$. We also relate $\chi<sup>c(G,k,r,p)$ to the number $|E|$ of hyperedges, and the maximum {\it hyperedge degree (dependency)} $d(G)$, as follows. We show that $\chi^{c(G,k,r,p)\leq} x$ for integer $x&gt;0$, if $$|E|\leq \frac{1}{2}({\frac{r^{k}{(t-1)^k} \binom{r}{t-1}}})^x$$, for any $k$-uniform hypergraph. We prove that a { \it strong $(r,p)$ cover} of size $x$ can be computed in randomized polynomial time if $$d(G)\leq \frac{1}{e}({\frac{r^{k}{(p-1)^k} \binom{r}{p-1}}})<sup>x-1$$.