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Concept

Cycle — where it appears

A closed walk in a network that returns to its start without reusing an edge; in a permutation, objects each sent to the next and the last back to the first. Cycles are what Euler's circuits are built from, and every permutation is a collection of disjoint ones.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

The odd ring that forbids a stable pairing. The ranked lists of 6 people and a stable partition of them drawn on a circle: pairs as plain chords, a ring of three or more as arrows from each person to the one they hold. An odd ring is present, as it is in every stable partition of this instance.

A ring that no pairing can break

Put everybody in one pool and a stable pairing may not exist. Allow rings as well as pairs and something stable always exists — and the pairs-only answer fails exactly when that stable arrangement contains a ring of odd length. Two sides make every ring even, which is the whole reason the two-sided theorem holds.

applied · Stable matching Hierholzer's construction on 6 vertices and 9 edges. Three views of one graph whose vertices all have even degree: a first closed walk that stops back at its start, the loops walked from vertices on it with edges left over, and the single circuit made by splicing them, with every edge numbered in order.

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.

discrete · Eulerian paths The cheapest way to send 4 units, for 25. A directed network from s to t with a capacity and a price on each road, showing the whole-number flow of 4 units with the least total cost, the arrows thickened by the amount they carry.

The cheapest way to send

Put a price on every road as well as a capacity and ask for the cheapest way to send four units. Twenty-eight ways exist and one is cheapest, and two certificates prove it without comparing it with the other twenty-seven: no cycle of roads it leaves unused costs less than nothing to push round, and there are prices at the places that every usable road fails to beat.

discrete · Network flow Five cycles of the Petersen graph that cover every edge twice. Petersen graph: 57 cycles; minimum cycle double cover has 5 cycles of lengths 5, 8, 6, 6, 5; 1654 search steps.

Every edge on exactly two cycles

Draw a graph on paper without crossings and its faces are cycles, each edge on exactly two of them. George Szekeres and Paul Seymour asked in the 1970s whether every graph without a bridge has such a set of cycles, drawn on paper or not. The Petersen graph needs five, the flower snark on twenty points five as well, every random graph tried has one, and nobody has proved it — though a minimal counterexample would have to be a snark.

discrete · Eulerian paths A function on nine points, and the tree with two marks it becomes. Left: a function on 9 points as arrows, with cycles through 2, 5, 7 and trees hanging into them. Right: the tree Joyal's rule makes from it, a path from head 7 to tail 2 with the same trees hanging below.

Every function is a tree with two marks

There are n to the n functions from n points to themselves, and n to the n − 2 trees on those points. André Joyal noticed in 1981 that the missing factor of n² is a choice of two points — a head and a tail — and that a function, read the right way, simply is a tree with a head and a tail. The reading turns Cayley's formula into one line and hands over a fact about random trees from the birthday problem.

discrete · Labelled trees Every pair of rearrangements of four letters, and what the two generate. A 24 by 24 grid of ordered pairs of permutations of four letters: 216 generate S4, 96 generate A4, 264 generate a smaller subgroup.

Two random shuffles reach every shuffle

Pick two ways of rearranging n letters at random and allow them to be repeated in any order. Almost always the two together reach every rearrangement there is, or every even one — and when they fail, the usual reason is a coincidence about a single letter that both happen to leave alone. Eugen Netto guessed it in 1882 and John Dixon proved it in 1969.

algebra · Symmetry groups

Named alongside it

The objects these essays reach for when they reach for this one.

Exhaustive searchParityPermutationExistence proofAlternating groupBijectionBirthday problemBlocking pairBridgeCapacityCertificateConjecture

All concepts