I need some words, some adjectives. Pretend for a moment that you are a coordinator for social events like speed dating. Your event will have some number of rounds. During each round, you will pair up the participants.

I need some adjectives to describe events. For example, you
might call an event *complete* if every potentially compatible
pair got paired up in some round of the event. I would prefer words
without negating prefixes when possible.

So, I need adjectives for the following concepts:

- Every potentially compatible pair was paired during some
round. (
*complete*,*covered*,*total*, ???) - No one was ever paired with the same person in more than
one round. (
*nonredundant*,*unrepeating*,*drive-by*,*passing*, ???) - Say that
**n**is the maximum number of people that any participant is potentially compatible with, and we manage to get together every compatible pair in only**n**rounds. (*speedy*,*efficient*,*fast*, ???) - No one ever has to sit out for a round. (
*engaging*,*efficient*,*non-idle*,*busy*, ???)

Let **P** be the set of participants. Let **C** be a
subset of the power set of **P**. Call the elements of **C**
the potentially compatible groupings. For a typical speed-dating
scenario, all potentially compatible groupings would be of size-two
(and, for a heterosexual event would consist of one male and one
female in each), but we're going to allow for more general
groupings.

Let **S** be a set of rounds **{ R _{1}, R_{2},
..., R_{k} }** where each round is a subset of the
power set of

**C**and we have

**∩ { g ∈ R**for all

_{j}} = ∅**j**. (The gist here is that no person can participate in two different groups in a single round.)

The above-mentioned situtations that I want adjectives for are more formally:

**∀ g ∈ C**there exists**R**with_{j}∈ S**g ∈ R**._{j}**g ∈ R**and_{i}**g ∈ R**if and only if_{j}**i = j**.- Condition #1 is true and
**| S | = max**_{x ∈ P}| { g ∈ C | x &isin g } | **∩ { &cup R**._{j}| R_{j}∈ S } = P

There are obviously a ton of combinatoric questions from here.
For example, given a simple graph **G**, let **C** be the set
of all pairs of vertexes which share an edge. Which of the above
adjectives can be realized for **G = K _{m,n}** or

**G = K**? Or, how many distinct

_{n}**S**are there for a given

**P**and

**C**?

I wouldn't be surprised at all to find there's already a body of mathematical literature most or all of this. But, I'm not sure at all what to search for. Any pointers would be much appreciated.