Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  introduce the notion of the { \it strong $(r,p)$ cover} number $\chi^c(G,k,r,p)$ for $k$-uniform hypergraphs $G(V,E)$, where $\chi^c(G,k,r,p)$ 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 $\chi^c(G,k,r,p)$ $r$-colorings.  derive the exact values of $\chi^c(K_n^k,k,r,p)$ for small values of $n$, $k$, $r$ and $p$, where $K_n^k$ denotes the complete $k$-uniform hypergraph of $n$ vertices.  study the variation of $\chi^c(G,k,r,p)$ with respect to changes in $k$, $r$, $p$ and $n$;  show that $\chi^c(G,k,r,p)$ is at least (i) $\chi^c(G,k,r-1,p-1)$, and, (ii) $\chi^c(G',k-1,r,p-1)$, where $G'$ is any $(n-1)$-vertex induced sub-hypergraph of $G$.  establish a general upper bound for $\chi^c(K_n^k,k,r,p)$ for complete $k$-uniform hypergraphs using a divide-and-conquer strategy for arbitrary values of $k$, $r$ and $p$.  also relate $\chi^c(G,k,r,p)$ to the number $|E|$ of hyperedges, and the maximum {\it hyperedge degree (dependency)} $d(G)$, as follows.  show that $\chi^c(G,k,r,p)\leq x$ for integer $x>0$, if $|E|\leq \frac{1}{2}({\frac{r^k}{(t-1)^k \binom{r}{t-1}}})^x $, for any $k$-uniform hypergraph.  prove that a { \it strong $(r,p)$ cover} of size $x$ can be computed in randomized polynomial time if $d(G)\leq \frac{1}{e}({\frac{r^k}{(p-1)^k \binom{r}{p-1}}})^x-1$.
['Tapas Kumar Mishra', 'Sudebkumar Prasant Pal']
I am writing a research paper on a idea Strong $(r,p)$ Cover for Hypergraphs
[{"from": "human", "value": "I am writing a research paper on a idea Strong $(r,p)$ Cover for Hypergraphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  introduce the notion of the { \\it strong $(r,p)$ cover} number $\\chi^c(G,k,r,p)$ for $k$-uniform hypergraphs $G(V,E)$, where $\\chi^c(G,k,r,p)$ 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 $\\chi^c(G,k,r,p)$ $r$-colorings.  derive the exact values of $\\chi^c(K_n^k,k,r,p)$ for small values of $n$, $k$, $r$ and $p$, where $K_n^k$ denotes the complete $k$-uniform hypergraph of $n$ vertices.  study the variation of $\\chi^c(G,k,r,p)$ with respect to changes in $k$, $r$, $p$ and $n$;  show that $\\chi^c(G,k,r,p)$ is at least (i) $\\chi^c(G,k,r-1,p-1)$, and, (ii) $\\chi^c(G',k-1,r,p-1)$, where $G'$ is any $(n-1)$-vertex induced sub-hypergraph of $G$.  establish a general upper bound for $\\chi^c(K_n^k,k,r,p)$ for complete $k$-uniform hypergraphs using a divide-and-conquer strategy for arbitrary values of $k$, $r$ and $p$.  also relate $\\chi^c(G,k,r,p)$ to the number $|E|$ of hyperedges, and the maximum {\\it hyperedge degree (dependency)} $d(G)$, as follows.  show that $\\chi^c(G,k,r,p)\\leq x$ for integer $x>0$, if $|E|\\leq \\frac{1}{2}({\\frac{r^k}{(t-1)^k \\binom{r}{t-1}}})^x $, for any $k$-uniform hypergraph.  prove that a { \\it strong $(r,p)$ cover} of size $x$ can be computed in randomized polynomial time if $d(G)\\leq \\frac{1}{e}({\\frac{r^k}{(p-1)^k \\binom{r}{p-1}}})^x-1$."}]