A matching that covers all 5 of one side
bipartite is one function. Everything below came out of it during this
build, at parameters taken from the essays rather than invented for this page — so a figure
here is the same figure a reader meets in an essay, and if the generator changes, this page
changes with it.
At its defaults
show: "hall"
show: "regular"
show: "deficiency"
What it checks while it draws
Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.
- round 1 still has a matching covering every vertex ×3
- and every neighbour named is a vertex on the right ×1
- and no vertex is matched twice ×1
- and so does every vertex on the right ×1
- and the reason is a set with too few neighbours between them ×1
- and the rounds use up every edge exactly once ×1
- between two and four graphs the family knows are compared ×1
- every edge gets exactly one colour ×1
- every matched pair is an edge of the graph ×1
- every vertex on the left has a neighbour list ×1
- every vertex on the left has the same degree ×1
- the graph is one the family knows ×1
- the largest matching is exactly the left side less the worst deficiency ×1
- the table has both a graph that succeeds and a graph that fails ×1
- the view is one the family draws ×1
- this view is for a graph that has no complete matching ×1
- this view is for a graph whose matching covers the whole left side ×1
Where it is called
Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.
A lottery over whole assignments
A table of shares in which every person's shares add to one task and every task is exactly covered is never anything more than a mixture of whole assignments — and finding the mixture is a matter of taking one complete assignment out at a time.
ComputationNine thousand four hundred and eight
There are four Latin squares of order four once the first row and column are fixed, fifty-six of order five, and nine thousand four hundred and eight of order six. The exact answer is known for eleven orders and for no more — and yet a half-finished square can always be finished.
DiscreteOne bottleneck and nothing else
A set of jobs can be filled by distinct people unless some group of jobs has too few candidates between them — and that single obstruction is the only one there is, which is what makes the theorem worth having.
DiscreteThe bottleneck is the whole story
However much a network can carry from one place to another, there is a way of cutting it in two whose total capacity is exactly that number. One quantity is a maximum over ways of routing and the other a minimum over ways of severing, and they are never off by even one.
DiscreteThe edge that forces a triangle
A graph on six points can carry nine edges with no three of them closing a triangle. It cannot carry ten. The bound is n²/4, the graphs that achieve it are all the same shape, and both facts fall out of examining every graph there is.
AlgebraThe same sum without its minus signs
Delete the signs from the determinant's sum over permutations and what is left counts things directly rather than by cancellation. It is a better count and a far worse object — because the cancellation was what made the determinant computable.