There’s a simple algebraic way to show that some graphs are nonplanar. If $G$ is a connected graph with $v$ vertices and $e$ edges, then it is nonplanar if $e > 3(v - 2)$. If it is also triangle-free, then it is nonplanar if $e > 2(v - 2)$. The first inequality shows that $K_5$ is nonplanar. The second shows that the utility graph is nonplanar. These are, essentially, the only “simple” nonplanar graphs. How many nonplanar graphs can actually be detected with this inequality?
We may answer this question using the wonderful
geng program, part of the
nauty distribution, which
generates graphs extremely quickly. I have written a short program which will
generate all connected graphs on a given number of vertices, pick out the
nonplanar ones, and see which fall to the above inequalities. This could tell
us roughly how surprised we should be when the simple inequalities work.
Let us call a nonplanar graph detectable if it satisfies $e > 3(v - 2)$, or $e > 2(v - 2)$ if it is triangle-free. The following table summarizes how many detectable graphs there are for $5 \leq v \leq 10$.
|Vertices||Connected nonplanar graphs||Detectable graphs||Proportion|
The only nonplanar graph with $v = 5$—$K_5$—is detectable, but after this examples are scarcer. Roughly 39% of the 13 nonplanar $v = 6$ graphs are detectable, and only 20% of the 207 $v = 7$ graphs are detectable. It would be nice to continue this table to see if the proportion column tends to any particular limit. However, this may be difficult.
The number of graphs with $v$ vertices grows very quickly. For $v = 6$ there are only 112 of them, but by $v = 10$ there are 11,716,571 such non-isomorphic graphs, or just under $12$ million. The computer that I computed this table on can run through about 2,000 graphs per second with $v = 10$. As expected, it took about an hour-and-a-half. According to the OEIS, there are 1,006,700,565 connected graphs with $v = 11$, or just over a billion. Generously assuming that we could still churn through 2,000 graphs a second, it would take about 139 hours to get an answer for $v = 11$, or just under 6 days.
However, we can extend it in one nice way. There are not many triangle-free graphs. Indeed, for $v = 11$, there are only 90,842 of them. We can crank through this is no time at all! Here is an updated table, using only triangle free graphs:
|Vertices||Triangle-free, connected nonplanar graphs||Detectable graphs||Proportion|
This still isn’t quite enough data to conclude anything.
I recently posted this question to the mathematics stack exchange. Surprisingly, my gut feelings were completely wrong! Through studying properties of random graphs, it turns out that we can say that “almost all” (in a probabilistic / counting sense) graphs are “obviously nonplanar,” in the sense that they violate the inequality $e \leq 3(v - 2)$. In particular, almost all graphs have average degree close to $n / 2$, while a graph with $e \leq 3(v - 2)$ has average degree a bit less than $6$.