In my graph theory class we've had several questions where we were to find the crossing number of a graph and prove our answer. In all of these questions, $Cr(G) = 1$, so we need only show a drawing of the graph with 1 crossing and that the graph is not planar either using one of the 2 common inequalities relating vertices ($q \leq 3p - 6$ for any planar graph $q \leq 2p - 4$ for a planar bipartite graph) or by edge and vertex contraction to find a resulting graph that contained either $K_{3,3}$ or $K_5$ (If either of these graphs were found, the graph is nonplanar by Kuratowski's Theorem).
I've come across a problem in my textbook where it asks the reader to prove that a graph has crossing number 5 and provides a picture (included below) that shows a drawing of the graph with exactly 5 crossings.
So, clearly $Cr(G) \leq 5$ given the drawing, but to complete the proof I must show that $Cr(G) \geq 5$, but I know only that I have been able to contract vertices and edges to find $K_{3,3}$ or $K_5 \implies Cr(G) \geq 1$. What should I do to show that the crossing number is $\geq 5$?
Or, more generally how to show that $Cr(G) \geq n, n \in \mathbb{N} - \{1\}$?
After further review I have found a proof as follows:
We have that $cr(G) \leq 5$ given the drawing presented. If $cr(G) < 5$ then removing 4 edges may create a planar graph, but $\forall$ planar graph we have $q \leq 3p - 6$. Given $p=10, q=29$ in the graph shown we would then have $29-4=25 \leq 3(10) - 6 = 24$, a contradiction. Thus $cr(G) \geq 5$. Then, necessarily, $cr(G) = 5$. $\square$