I get some obviously unwanted space here:
Does it have to do with invisible control points being used to draw the arcs?
Here's the offending code:
\begin{tabular}{|c|c|}
\hline
\begin{tikzpicture} [main/.style = {draw, circle},node distance = 2cm,on grid,auto]
\node[main] (1) {$1$};
\node[main] (2) [right of=1] {$2$};
\node[main] (3) [below of =2] {$3$};
\node[main] (4) [left of =3] {$4$};
\draw (1) -- (2);
\draw (2) -- (3);
\draw (3) -- (4);
\draw (4) -- (1);
\draw (1) -- (3);
\draw (2) to [out=135,in=180,looseness=2.5] (4);
\end{tikzpicture}
&
\begin{tikzpicture} [main/.style = {draw, circle},node distance = 2cm,on grid,auto]
\node[main] (1) {$1$};
\node[main] (2) [below right of= 1] {$2$};
\node[main] (3) [below left of =1] {$3$};
\node[main] (4) [below of =3] {$4$};
\node[main] (5) [below of =2] {$5$};
\draw (1) -- (2);
\draw (1) -- (3);
\draw (1) -- (4);
\draw (1) -- (5);
\draw (2) to [out=90,in=90,looseness=2.5] (3);
\draw (2) to [out=-50,in=-50,looseness=2.5] (4);
\draw (2) -- (5);
\draw (3) -- (4);
\draw (3) -- (5);
\draw (4) -- (5);
\end{tikzpicture}
\\ \hline
$K_{4}$ drawn without crossing & Attempted drawing of $K_{5}$ without crossings \\ \hline
\end{tabular}

