Mathematical Proof that Yuri is a Realm of Endless Possibilities

Well, exponentially greater possibilities, at least.

One of the great things about yuri (or yaoi, or, even better, bisexual characters) is that the number of possible shippings increases exponentially more with the number of characters. Let’s look at why yuri is awesome and learn a bit of graph theory while we’re at it!

First of all, how do we define a “shipping”? Generally, the term refers to a single couple: for example, I have been shipping Ringo and Shouma in Mawaru Penguindrum. However, people can ship multiple couples in a single show: for our purposes, we’ll refer to this set of pairings as a single shipping. To simplify things, we’ll assume that every character is matched to a single partner. So no threesomes or harem end.

Love Graphs

To count how many possible pairings a show has, we’re going to construct a “love graph.” Let’s start with the basics: what is a graph? The graph G = (V, E) is a set of vertices V and a set of edges E. An edge links two vertices in the graph. So in the love graph, each character is a vertex, and edges represent possible romantic links. So (a subgraph of) the love graph for Kimikiss would look like this:

We have the love triangle surrounding Kazuki, Kouichi’s love triangle, and Eiji as Kouichi’s rival for Mai. Note that the graph is bipartite: we can divide the vertices into two sets, such that all edges in the graph cross from one set to the other, simply by picking a set of boys and a set of girls. The pairings in Kimikiss are all straight.

The number of possible shippings, then, is the number of matchings on the love graph. A matching is simply a set of edges which share no common vertex— a set of couples where nobody is two-timing.

Of course, the pairings shown on the above graph are only the ones that the show itself encourages. What if we think that Sakino and Sanada make the perfect couple? Nothing wrong with that (except that Sanada is an asshole for choosing Mai and deserves to die alone and unloved). So the graph we would consider for Kimikiss in counting the number of possible shippings has edges between every pair of characters of the opposite sex. If we permit bisexual pairings, then the love graph becomes complete: there is an edge from every character to every other character in the graph.


Your typical harem has a single male character of consequence, and n females who are interested in him. How would this love graph look?

It would also be bipartite: the male character is linked to every female character, and there are no links between the female characters. If the graph has n females, then, the love graph has n possible matchings, and there are n possible shippings. Note that no perfect matching exists in the harem love graph: n-1 females must die alone and unloved. This is their lot, and we shall shed no tears for them.

Heterosexual Romance

Now let’s consider shows like Kimikiss and ef, where we have a similar number of males and females. Assume there are n characters, with an equal number of males and females. In this case, a perfect matching exists: to simplify things, we’ll count only the number of perfect matchings, where every character is part of a couple. Once again, the love graph is bipartite, with two fully interconnected vertex sets of size \frac{n}{2}. There are \left(\frac{n}{2}\right)! such pairings. (To see this, we have \frac{n}{2} females to match the first male to, \frac{n}{2} - 1 females to match the second male to, etc. )

Yuri / Yaoi / Bisexual

Next, consider a yuri / yaoi / bisexual show such as Yuruyuri or Simoun, with a cast of n characters, any two of which are a potential item. Now our love graph is complete: every character can form a couple with every other character. Once again, we’ll count the number of perfect matchings. Assuming n is even (which it must be for a perfect matching to exist) we can determine that the number of perfect matchings is \phi(n) = (n-1)(n-3)\cdots3 \cdot 1 using a similar argument to last time.


To summarize:

So, the number of characters being equal, a show where the genders are evenly distributed has \frac{1}{2}\left(\frac{n}{2}-1\right)! times as many perfect (well, for one gender) matchings as a standard harem . And yuri shows have

\frac{\phi(n)}{\left(\frac{n}{2}\right)!} = \frac{1}{\frac{n}{2}} \prod_{i=1}^{n/2 -1} \frac{2i+1}{i} \ge \frac{2}{n} \prod_{i=1}^{n/2-1} 2 = \frac{2}{n} 2^{n/2-1} = \frac{1}{n} 2^{n/2}

times as many possible perfect matchings as a heterosexual show.

So there we have it. Yuri is awesome, a realm of exponentially greater possibilities. Q.E.D.

Liked this post? Leave a comment, subscribe to our RSS feed, and follow us on Twitter!

11 thoughts on “Mathematical Proof that Yuri is a Realm of Endless Possibilities

  1. Your post title is more selective than your deductions. That is, in the title you put “yuri” over more mainstream possibilities, and you prove that in the post, but even though you also prove the superiority of “yaoi” that genre is missing from both the title and your concluding sentence. So now my question is: is there an OBJECTIVE method for comparing the relative potentials of “yuri” and “yaoi”, or are they both the same? My sense right now is that, for reasons of anatomy, “yaoi” might present even more possibilities for “pairing up” than “yuri”.

    1. The title is simply because I prefer yuri. That part is purely subjective. 🙂

      The number of pairings if we count in terms of couples is the same for both yaoi and yuri. If we consider different anatomical arrangements, my intuition is that you are correct, although I am not an expert in this field. If so, the number of possibilities for “pairing up” would be a constant factor greater for yaoi. The bisexual case depends on the proportions of male and female characters, but would fall between the two extremes.

      I’m a computer scientist though, so I would say that asymptotically yaoi and yuri have the same behavior; they only differ by a constant factor. The difference is minor when compared with heteronormative shows.

  2. Subjective me: Awesome article! Indeed, “yuri is awesome, a realm of exponentially greater possibilities.”

    Objective me: I have trouble with your (n-1)(n-3)…3.1 permutation. How I understand is:

    Assuming that n=even with equal number of females and males. I think it should be
    bi- permutation = (n-1)! -> pairings which include everything because each go both sides
    yuri/yaoi permutation = ((n/2)-1)! -> pairs from half of the set set only

    Correct me if I’m wrong. But overall, I enjoyed your idea. I didn’t know you’re into yuri.

    1. Ah, I didn’t spell this out too clearly, but for the yuri case we’re dealing with n people, all of the same gender, rather than half and half. (since generally yuri shows will have only / mostly lesbians) There aren’t n! perfect matchings though because we’re drawing people from the same set rather than two different sets like we did before.

      So to count the perfect matchings over the set of n people, first fix one person, choose her partner. There are n-1 possibilities. Now there are n-2 unassigned people left. Fix one, choose her partner, there are n-3 possibilities. And so on and so forth. So there are (n-1)(n-3)…*3*1 possible perfect matchings.

      And who isn’t into yuri? Simoun is my favorite show, although it’s more bisexual.

  3. Can’t argue with your logic, but something is telling me you must been watching a bunch of yuri lately. Anyway, brilliant post, since I do concur that yuri offers a myriad of paths. Too bad I still refuse to pick up yuri themed visual novels/eroge. Just not my cup of tea…

    1. I don’t read yuri visual novels / eroge either… well, more like I don’t read visual novels at all… BUT, yes, I actually wrote this last season while I was watching Yuruyuri. Great show.

  4. Endless possibilities, but not many yuri anime series…I’ve heard of Aoi Hana, but that looks like a different type of show that YuruYuri…

    Any Yuri manga recommendations?

    1. There aren’t many comedies like YuruYuri, unfortunately. There are a decent amount of more serious yuri anime though: Aoi Hana, Sasameki Koto, Strawberry Panic, MariMite, Kashimashi, Kannazuki no Miko and Simoun, to name a few.

      I don’t read much manga unfortunately. I’ve heard great things about this manga called “Girlfriends” and it’s on my to read list, so you might want to give that a try.

  5. Your logic is irrefutable. I love this post.

    Exponential growth possibility for yuri, indeed! Now, let’s see if we can somehow make this into a proven widely accepted statement in anime…

Leave a Reply

Your email address will not be published. Required fields are marked *