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Does anyone know the answer or bounds for: The number of distinct Maximal Planar Graphs, with $n$ vertices? Just simple graphs please, without loops or directions.

It's because there is an efficient algorithm which $4$ colours planar graphs in typically $\log _{n}$ attempts. The algorithm could also enumerate the number of graphs for small $n$, but slows down for large $n$. Is the number already known?

RobPratt
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John Hunter
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  • Welcome to MSE! Please use the basic tutorial and quick reference guide and also show the work you have done so far. – Jessie Feb 02 '21 at 14:27
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    Searching for "maximal planar graphs" in the Online Encyclopedia of Integer Sequences yields A000109. The OEIS indicates that this sequence is "hard". – Will Orrick Feb 02 '21 at 16:11
  • I've just spent far too much time figuring out the result for 6: it's 1, namely $K_6$ minus the order-4 path graph. (there are three minimal non-planar graphs of this size: $K_5$ plus a point, $K_3,3$, and $K_5$ but with one of the edges subdivided) – Dan Uznanski Feb 02 '21 at 16:44
  • @DanUznanski What about the octahedron? – Will Orrick Feb 02 '21 at 17:39
  • Argh, I must have erred at some point. I had that marked as non-planar for some reason :(. Okay it is two. – Dan Uznanski Feb 02 '21 at 17:48
  • @DanUznanski Two agrees with the OEIS sequence. I haven't checked myself though. – Will Orrick Feb 02 '21 at 18:06
  • Thanks, the A000109 list is interesting, is there a known formula for that list? Perhaps it's related to the A000108 numbers, the Catalan numbers as follows...to generate Maximal Planar graphs the algorithm did the following... – John Hunter Feb 02 '21 at 21:14
  • It seems that every Maximal Planar graph has this representation. Start with a convex polygon e.g. a hexagon and triangulate the inside with 'inner connections'. Also triangulate in a similar way on the outside, or inside again if you prefer with dotted lines. The number of edges is n for the polygon, n-3 for the inner connections and n-3 for the outer connections, giving 3n-6. Care is needed not to repeat a connection. Perhaps the number of distinct Maximal Planar Graphs could be got this way, a weak upper limit is Cn-2 squared. There are Cn-2 ways to triangulate the inside and outside. – John Hunter Feb 02 '21 at 21:21
  • Generally an OEIS entry will have a "Formula" section if a formula is known, or a "Program" section if there is an algorithm that can be written down in a few lines. But A000109 has neither. Did you look at the linked paper by Brinkmann and McKay? This sequence appears in Table 1 and the paper describes the software used to compute the sequence. (The software can be downloaded.) – Will Orrick Feb 02 '21 at 22:49
  • Thankyou, do you think the C(n-2) squared upper limit can be reduced? Rotations for the initial 'inner' connections divide it by about n...there must be lots of reflections too, before even adding the 'outer connections', then there must be lots of duplications of the graphs. I'll have a look at their program, but a formula for the bounds would be interesting too. – John Hunter Feb 02 '21 at 23:09
  • By starting with an $n$-gon and then triangulating aren't you assuming that every maximal planar graph is Hamiltonian? I don't think that's true. – Will Orrick Feb 03 '21 at 00:28
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    Example: Goldner–Harary graph – Will Orrick Feb 03 '21 at 04:49
  • Thankyou, the counter example is good. That leaves us with looking for bounds on the number of distinct Maximal Planar Graphs, the A000109 numbers. Presumably the non-Hamiltonian are rare compared to the Hamiltonian ones. They seems to follow C(n-2)*2^(n-13) quite closely, but go greater gradually, where C(n) are the Catalan numbers A000108, which are also the number of ways to divide a polygon...any ideas about bounds anyone? – John Hunter Feb 03 '21 at 09:17
  • It might be worth editing your question to incorporate some of the information in the comments. This could help draw attention to the issues you are interested in, as not everyone reads all the comments. My impression is that this is a difficult instance of a class of difficult problems about which not much is known. – Will Orrick Feb 03 '21 at 15:20
  • Thankyou, it's probably as far as we can go with this at the moment, just curious if there was a formula, or known bounds. All the best. – John Hunter Feb 03 '21 at 18:59
  • Think a new questions will be started along the lines of - 'How many distinct Maximal Planar Graphs can be formed like this... triangulating a regular a polygon inside and on the outside to form a Maximal Planar Graph'. (simple graphs, no loops, repeated edges directions). Perhaps it will give an idea for a formula or bounds for the A000109 numbers, presuming that most Maximal Planar Graphs can be represented this way...any comments before it's posted? – John Hunter Feb 04 '21 at 22:50
  • Seems reasonable. I don't have a feeling for whether the Hamiltonian graphs remain predominant or not. Two papers I neglected to link to earlier: Chen, Helden – Will Orrick Feb 05 '21 at 00:05
  • Also relevant: "Triangulations with few ears: symmetry classes and disjointness" by A. Asinowski and A. Regev. – Will Orrick Feb 05 '21 at 03:12
  • Thankyou, it'll be posted today or tomorrow – John Hunter Feb 05 '21 at 08:45
  • Link to the new question. Also, I've left a comment there suggesting that the non-Hamiltonian graphs may not be rare after all once the graphs get large. – Will Orrick Feb 20 '21 at 07:40

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Despite all the back-and-forth in the comments and in the follow-up posts, I managed to overlook something important. There is a simple formula for the number of labeled maximal planar graphs, which is a result of Tutte. From this, one gets an immediate lower bound on the number of maximal planar graphs. And assuming an extremely plausible hypothesis, this lower bound becomes tight for large $n$.

Update: The hypothesis turns out to be a theorem, which makes the asymptotic result a theorem as well. (See below.)

In the paper

W. T. Tutte, A census of planar triangulations, Canad. J. Math 14 (1962) 21–38,

Tutte derives a result implying the the number of labeled maximal planar graphs of $n$ vertices (OEIS entry A007816) is $$ \binom{n}{2}\frac{(4n-11)!}{(3n-6)!}. $$ (If I can find a straightforward derivation of this, I will add it to the post.) If all labeled maximal planar graphs had trivial automorphism group, the number of maximal planar graphs would simply be this expression divided by $n!$. Because some labeled maximal planar graphs have an automorphism group of order greater than $1$, however, those graphs get a larger weight than do those with automorphism group of order $1$. As a consequence, a lower bound on the number of maximal planar graphs is $$ \left\lceil\frac{1}{n!}\binom{n}{2}\frac{(4n-11)!}{(3n-6)!}\right\rceil. $$

Comparing this lower bound with the exact values given by OEIS entry A000109, one gets $$ \begin{array}{r|r|r} n & \text{lower bound} & \text{exact}\\ \hline 4 & 1 & 1\\ 5 & 1 & 1\\ 6 & 1 & 2\\ 7 & 2 & 5\\ 8 & 6 & 14\\ 9 & 31 & 50\\ 10 & 177 & 233\\ 11 & 1099 & 1249\\ 12 & 7150 & 7595\\ 13 & 48257 & 49566\\ 14 & 335668 & 339722\\ 15 & 2394471 & 2406841\\ 16 & 17450493 & 17490241\\ 17 & 129539473 & 129664753\\ 18 & 977116410 & 977526957\\ 19 & 7474578281 & 7475907149\\ 20 & 57891920643 & 57896349553\\ 21 & 453367652518 & 453382272049\\ 22 & 3585804266282 & 3585853662949\\ 23 & 28615537688289 & 28615703421545 \end{array} $$ One sees that for $n=23$ the lower bound and the exact value agree to five significant figures. This is not too surprising: one expects that as $n$ gets large, a graph is almost certain to have automorphism group of order $1$. In other words, one expects that a large graph will almost certainly have no non-trivial symmetry. I know of no proof of this, however, so we can take is as a hypothesis.

Assuming the hypothesis, Tutte converted the bound to an asymptotic estimate for the number of maximal planar graphs: $$ \frac{1}{64\sqrt{6\pi}}n^{-7/2}\left(\frac{256}{27}\right)^{n-2}. $$

Update: Tutte later rigorously proved the hypothesis, which makes the asymptotic estimate a theorem as well. See

W. T. Tutte, On the enumeration of convex polyhedra, J. Combinatorial Theory, Series B 28 (1980) 105–126,

Will Orrick
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