Stitching Graphs andPainting Mazes:

Problems inGeneralizations of

Eulerian Walks

Joshua Holden

Joint work with (anddiagrams by) Lana Holden

http://www.rose-hulman.edu/~holden

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Blackwork, also known as “Spanish stitch”, becamepopular in England around 1501.

Supposedly Catherine of Aragon brought it to England then.But in fact it was already known there and many other places.

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Blackwork is also known as “Holbein stitch”, thanks toHans Holbein the Younger, Henry VIII’s court painter.

His paintings are so detailed you can clearly see the stitching.3 / 32

The rules of blackwork embroidery depend(somewhat) on who you ask.

The most common materials traditionally are black thread andlight-colored linen.

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The stitching is traditionally done with “double runningstitch”.

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Also sometimes back stitch, but we will be talking about doublerunning stitch.

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The stitching is traditionally done with “double runningstitch”.

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Also sometimes back stitch, but we will be talking about doublerunning stitch.

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The stitching is traditionally done with “double runningstitch”.

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Also sometimes back stitch, but we will be talking about doublerunning stitch.

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Also, we will only be talking about reversible patterns,which appear the same from both sides of the fabric.

(“Betsy”, by Catherine Strickler, published by Indigo Rose)Reversible patterns were often used for cuffs and collars.

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A digraph is a set of vertices, V , and a set of edges, E ,where each edge is an ordered pair of distinct vertices.

The order is thought of as indicating a “direction”.7 / 32

An (undirected) graph may be associated to a digraphby forgetting about the ordering of the pairs.

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A (symmetric) digraph may be associated to a graphby including both possible directions of each edge.

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A walk on a graph is a finite alternating sequence ofvertices and edges x0, {x0, x1}, x1, . . . , {xn−1, xn}, xn.

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A graph is connected if there is a walk between anytwo vertices.

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A directed walk on a digraph is a finite sequence ofvertices and edges x0, (x0, x1), x1, . . . , (xn−1, xn), xn.

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A Eulerian circuit on a digraph is a directed walk withevery edge used exactly once and x0 = xn.

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Every symmetric digraph associated to a connectedgraph is Eulerian, i.e., has an Eulerian circuit.

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Having an Eulerian digraph is necessary for a patternto be reversibly stitchable, but is it sufficient?

The two different directions have to lie on opposite sides of the fabric.15 / 32

Having an Eulerian digraph is necessary for a patternto be reversibly stitchable, but is it sufficient?

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The two different directions have to lie on opposite sides of the fabric.15 / 32

If x0,e1, x1, . . . ,en, xn is a directed trail on a digraph,we say that the parity of each edge ei is the parity of i .

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A Holbeinian circuit on a digraph is an Eulerian circuitwhere all (x , y) and (y , x) have opposite parities.

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Theorem: Every symmetric digraph associated to aconnected graph is Holbeinian.

This will not be a surprise to any embroiderers.

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Theorem: Every symmetric digraph associated to aconnected graph is Holbeinian.

There are several ways we could prove this theorem.

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One way is to give an algorithm which produces thecircuit. This one is “Tarry’s Algorithm” (with parity).

Suppose we have a strongly connected symmetric digraph.1. Start at an arbitrary vertex x0.2. Proceed along any edge.

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One way is to give an algorithm which produces thecircuit. This one is “Tarry’s Algorithm” (with parity).

3. At each later step, suppose we have arrived at a vertex y .If y is not x0, let (x1, y) be the edge that first reached y(“entry edge”).

a. If there is an edge (y , z) other than (y , x1) which has notbeen traversed (and such that (z, y) either has not beentraversed or was traversed with parity opposite the currentparity) proceed along any such edge.

b. If every edge (y , z) other than (y , x1) has been traversed(or (z, y) was traversed with the current parity), leave along(y , x1) (“reverse of entry edge”) (if (x1, y) was traversed withparity opposite the current parity).

c. If there are no allowed moves as above, terminate thealgorithm.

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Theorem: Tarry’s Algorithm with parity alwaysproduces a Holbeinian circuit.

Example

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(False) Conjecture: Every Holbeinian circuit can beproduced from Tarry’s algorithm.

Counterexample

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(False) Conjecture: Every Holbeinian circuit can beproduced from Tarry’s algorithm.

Counterexample

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Fact: It is not possible for a “local” algorithm to tellwhether it is producing an Eulerian circuit or not.

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Fact: It is not possible for a “local” algorithm to tellwhether it is producing an Eulerian circuit or not.

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Conjecture: There is a “local” algorithm with an“oracle” which produces every Holbeinian circuit.

What sort of oracle?24 / 32

What if you would like to stitch a pattern so that the topand bottom threads go in the same direction?

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A multigraph of multiplicity 2, or 2-multigraph, is an(undirected) multigraph where each edge appearsexactly twice.

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An Aragonian circuit of a 2-multigraph is an Euleriancircuit where the two edges {x , y} are traversed in thesame order but with opposite parities.

(Same direction, opposite sides of the fabric.)

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Theorem: A multigraph of multiplicity 2 is Aragonian ifand only if the associated graph is Eulerian and has acircuit of odd length.

Example

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Theorem: A multigraph of multiplicity 2 is Aragonian ifand only if the associated graph is Eulerian and has acircuit of odd length.

Non-example

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What if some of the edges need to be stitched in a“Holbeinian way” and some in an “Aragonian way”?

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There are lots of other questions to be asked!

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Thanks, and happy stitching![1] Joshua Holden, The Graph Theory of Blackwork Embroidery, Making

Mathematics with Needlework (sara-marie belcastro and CarolynYackel, eds.), A K Peters, 2007, pp. 136–153.

A modern blackwork pattern, by the author(Title page model stitched by Ann Black)

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