\mychapter{Future Work}
\label{Future}

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As mentioned earlier, finding saturation numbers is hard in general.  A great deal of time and effort has been spend in finding saturation numbers for cycles.  Still the only exact values known are for $C_3, C_4$, and $C_5$.  There are also numerous other graphs whose saturation number is not known.  Then the next question is finding the edge spectrum for those graphs.  Here are some questions that can be considered next.


\begin{enumerate}
\item Edge spectrum for book and generalized book graphs.


\item Edge spectrum for paths.


\item Edge spectrum for a star.


\item Edge spectrum for complete bipartite graphs.


\item Edge spectrum for complete $t$-partite graphs.


\item Saturation numbers for cycles $C_t$, $t >5$.  


\item Edge spectrum for cycles.

\end{enumerate}


Recently, D. West and P. Wenger (personal communcation) defined saturation unique graphs to be the ones that are $H$-saturated but the addition of any edge produces exactly one copy of $H$.  Hence, the next natural question would be to find saturation unique graphs for complete graphs, bipartite graphs, paths, cycles, etc, if they exist.  Furthermore, let $f(k)$ be the minimum number of copies of $H$ in a $H$-saturated graph with $k$ edges after we add any edge.  Then finding this $f(k)$ for complete graphs, complete bipartite graphs, paths, cycles, etc., can be another future project.


