Graph — where it appears
Named by 41 essays across 9 fields — each of them below, with the objects they name alongside it.
Seven bridges, and the invention of throwing things away
Euler solved a puzzle about a Prussian city by deleting the city. What survived the deletion was a new branch of mathematics.
Four colours, and a proof nobody can read
Every map on a plane can be coloured with four colours so that no two neighbours match. The statement is understandable by a child, it resisted a century of attempts, and the proof that settled it cannot be checked by a human being.
Every corner pays for itself
Count the corners of any solid, subtract the edges, add the faces. The answer is two. It is two for a cube, for a pyramid, for a football, for anything squashed or stretched — and the number is measuring the shape it is wrapped around rather than the shape itself.
Six people at a party
Among any six people, three are mutual acquaintances or three are mutual strangers. Five is not enough, and the arrangement that saves five is a pentagon. Beyond that the numbers become unknowable.
The majority that goes in a circle
Every voter hands in a ranking, and a ranking is transitive by construction. Compare the candidates two at a time and let the majority decide each pair, and the verdicts need not fit together into a ranking at all.
The road that makes everyone later
An equilibrium is a state nobody can improve alone, which is a much weaker thing than a state anybody would choose. Adding a link that costs nothing to use makes every traveller in this network strictly slower, and the arithmetic says by exactly how much.
A walk that changes one thing at a time
Counting from nothing to fifteen in binary changes four digits at once somewhere in the middle. There is another order through the same sixteen words in which every step changes exactly one — and it is a closed walk on a four-dimensional cube.
Two graphs that will not lie flat
Five points, every pair joined: no matter how the points are placed or how the lines are drawn, two of the lines cross. The proof is not about drawing at all — it counts edges against faces and finds one edge too many.
Sixteen trees on four points
How many ways are there to connect n labelled points into a single tree? The answer is n to the power n minus two, which is a strange enough formula to demand an explanation — and the explanation is a code that turns every tree into a short list of numbers, and every short list of numbers back into a tree.
One 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.
Every word once, around a cycle
A cyclic string of eight bits holds all eight three-bit words, each exactly once — and the reason such a thing exists is that the constraint linking overlapping windows is itself the construction.
The subgroup that is freer than the group
A free group on two letters contains a subgroup of index three that is free on four. Nothing about a group makes that plausible; everything about a graph makes it obvious, and the argument is to stop looking at the group and start looking at the space whose loops it is.
Eighteen people, and the seventeen that escape
Among any eighteen people, four are mutual acquaintances or four are mutual strangers. Seventeen can be arranged so that neither happens, and the arrangement is not a lucky find — it is a rule about squares.
A determinant that counts trees
Write down a graph's Laplacian, strike out one row and its column, take the determinant. The answer is the number of spanning trees — and the minus signs in the determinant are what cancel every subset of edges that is not one.
A page that knows where it is
A four-by-four array of bits, cyclic in both directions, in which every two-by-two block appears exactly once. Print it repeatedly across a sheet and any four marks on that sheet are an address.
A cycle for every pair
A cyclic sequence in which every window of two consecutive symbols is a different pair of things. For five things it exists and for four it does not, and in both cases there are exactly as many pairs as there are places to put them.
Every flat graph is a pile of circles
A graph that can be drawn without crossings can be drawn in one particular way: as circles, one per vertex, touching exactly when their vertices are joined. The picture is not a choice — it is determined, up to the group two inversions generate.
The chain that runs the same backwards
Put weights on the edges of a graph, step to a neighbour in proportion to them, and the long-run share of a state is its own weight over the total — read straight off the picture, with nothing to solve. The condition that makes that work is strictly stronger than being stationary.
The tree inside the triangulation
The shortest network joining a set of points is built from edges chosen by length, and the triangulation is built from edges chosen by an emptiness condition about circles. The two constructions share no step, and every edge of the first is an edge of the second.
The solid whose corners are triangulations
Take the triangulations of a hexagon as points and join two of them when a single diagonal can be swapped for another. The result is not merely a graph — it is the edge skeleton of a genuine convex polyhedron, with fourteen corners, three square faces and six pentagonal ones.
The landscape nobody is looking at
Letting participants move one at a time to whatever is currently better can cycle forever, and on a network of congestible roads it cannot. The reason is a single number attached to each state that falls by exactly what the mover saves.
The symmetries a cover has of its own
A covering space can be shuffled without disturbing anything below it, and how many ways there are is decided by the subgroup it corresponds to. When there are as many symmetries as sheets the covering is called regular, and that is the same statement as the subgroup being normal.
Folding a graph until it decides
A subgroup of a free group usually arrives as a list of words, and almost nothing about it is readable from the list. Draw the words as loops, merge every pair of edges with the same label leaving one point, and what is left is a machine that decides membership by reading.
One tree for every cut
A network of six places has fifteen pairs, and each pair has its own cheapest cut. All fifteen can be read off a tree with five numbers on it: the cheapest cut between any two places is the smallest number on the tree's path between them. Gomory and Hu proved in 1961 that such a tree always exists, and building it takes five cuts, not fifteen.
Two literals make an arrow
A clause of two literals, p ∨ q, says that if p is false then q is true, and if q is false then p is true — two arrows. A set of such clauses is a directed graph on the literals, and it is unsatisfiable exactly when some variable and its negation reach each other. Each of those two paths is a chain of resolution steps, and finding them takes time proportional to the size of the input.
What the search has when it fails
A largest matching is easy to find and hard to certify: the claim that nothing larger exists is a claim about every arrangement not tried. The certificate turns out to be free — it is the wreckage of the search that failed.
The piece that cannot pair off
Take the sides away and the obstruction to a matching changes character completely. It is no longer a shortage of partners; it is a parity, and the quantity that measures it counts pieces of odd size rather than vertices of any size.
The cycles and the cuts
The count that splits a map's source into what dies and what survives has nothing to do with graphs. Apply it to a matrix built from a graph's edges and points and it says that a graph's independent cycles and its independent cuts add to its number of edges — a theorem about drawings, obtained from an array.
The few points that cut a flat graph
Any graph that can be drawn without crossings, however large, falls into pieces of at most two thirds once a few points are removed — about the square root of its size, never more than 2.83 times it. A grid shows the square root cannot be beaten, a ring of breadth-first neighbours comes close, and a cycle through a shallow tree finishes the job.
Moves that only ever add edges
Turán's theorem says the densest graph avoiding a complete graph on r + 1 points is the balanced r-part graph. Zykov's proof finds it by a sequence of moves — turn a point into a copy of a better-connected one it is not joined to — each of which adds edges and none of which can create the forbidden clique. A second proof spreads a unit of weight over the points and gets the same bound from a maximum.
A walk that splices in its own detours
Euler proved that a walk crossing every bridge once needs every landmass to have an even number of bridges, and then stated, without proof, that this was enough. The missing half took 137 years, and it is not an argument but a procedure: walk until stuck, notice that stuck can only mean home, and splice in a detour from anywhere with edges left. The procedure never fails, and the reason fits in one sentence about arriving and leaving.
The streets a postman walks twice
A postman must walk every street of a district and come back. If every corner has an even number of streets, no street needs walking twice. If not, some must — and the ones repeated always join the odd corners in pairs. Pricing every way of pairing them finds the shortest round; pairing the nearest corners first does not.
Three places cut apart
Separating two places as cheaply as possible is solved exactly by a flow. Separating three from one another is a different problem: no flow measures it, the pairwise answers do not add up to it, and the best shortcut known in 1994 — cut each place off on its own and throw the dearest cut away — is guaranteed only to within a third of the truth.
The narrowest door sets the pace
How fast a chain forgets is an eigenvalue, and nobody can compute the eigenvalues of a chain worth studying. Cheeger's inequality trades the eigenvalue for a picture — the narrowest door in the state space — and pins the one between the square of the other and twice it. Both ends of that range are reached, on graphs small enough to search completely.
Every surface is sewn from pants
A sphere with three holes — a pair of pants — is the smallest piece a surface with two or more handles can be cut into. Every such surface falls apart into them, and the Euler characteristic alone says how many pieces and how many cuts, however the cutting is done. What it does not say is the pattern, and the patterns are counted by drawing each one as a graph: two for two handles, five for three, seventeen for four.
Two graphs the eigenvalues cannot tell apart
A graph's matrix has eigenvalues, and they count a surprising amount of the drawing: its edges, its triangles, every closed walk of every length. They do not count everything. A star with four arms and a square beside a lone point have the same eigenvalues exactly, although one of them is in two pieces — and on six points ten of the 156 graphs have a twin of this kind.
Every ordering once, around a cycle
No cycle can show every ordering of three symbols as a window of three: a window holding each symbol once forces the next symbol to repeat the one just dropped, so the sequence has period three and shows three orderings of six. Two repairs work. Write each ordering by its first two entries and the transitions form a balanced graph, so Euler's theorem hands over the cycle at once. Or add a fourth symbol and ask only that each window keep a different relative order — which works too, but no graph explains why.
How far apart two triangulations can be
Any triangulation of a polygon can be turned into any other by flipping one diagonal at a time. On an octagon seven flips always suffice; on a polygon with m vertices the worst case is 2m − 10 flips once m reaches thirteen, a value Sleator, Tarjan and Thurston proved with hyperbolic geometry in 1988. Breadth-first search over all 58,786 triangulations of the 13-gon finds the sixteen.
Resolving one letter away at a time
Davis and Putnam's procedure of 1960 decides a set of clauses by removing its letters one at a time, replacing every clause that mentions a letter by all the ways of resolving it away. On a cycle the clauses never multiply; on a grid they swell by a factor of three and a half for each extra row, and on the 4 × 4 grid each extra unit of the order's width doubles the peak exactly. The cost is not the size of the formula but how wide it is.
A walk on Gaussian primes stopped by a moat
Start at 1 + i and step from Gaussian prime to Gaussian prime, never more than 2 at a time: the walk reaches 720 primes and stops at 17 + 42i, surrounded by a moat. Allow steps of √10 and it reaches a quarter of a million primes before stopping at distance 1,024; steps of 4 carry it to 4,313. Whether some fixed step lets a walker reach infinity is a question Basil Gordon asked in 1962, and it is open.
The order that keeps elimination narrow
Eliminating letters one at a time costs whatever the widest step costs, and the least width any order can achieve is the treewidth of the formula's graph. Finding it exactly is NP-hard, but a search over sets of letters instead of sequences finds it for small graphs, and a greedy rule that adds the fewest new edges finds it on 566 of 600 random graphs and misses by more than one only once. On Tseitin's 6 × 6 grid that rule forms twenty-four times fewer resolvents than sweeping row by row. What it cannot do is prove that its order is the best.
Named alongside it
The objects these essays reach for when they reach for this one.
Counting argumentExhaustive searchExistence proofParitySpanning treePlanar graphComplete graphCounterexampleDe bruijn sequenceLower boundMatchingMatrix