An Eulerian walk traverses each edge of a graph exactly once. What happens if you want to traverse each edge of a graph exactly twice? If you want to cover the graph with "double-running stitch", then you need to traverse each edge twice but also put conditions on how many edges you traverse in-between. Then you could add conditions on whether you traverse the edges once in each direction or twice in the same direction. Which graphs can you still traverse? Classical algorithms for solving mazes give us some answers to these questions, but others are still open.

1. Stitching Graphs and
Painting Mazes:
Problems in
Generalizations of
Eulerian Walks
Joshua Holden
Joint work with (and
diagrams by) Lana Holden
http://www.rose-hulman.edu/
~holden
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2. Blackwork, also known as “Spanish stitch”, became
popular 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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3. Blackwork is also known as “Holbein stitch”, thanks to
Hans Holbein the Younger, Henry VIII’s court painter.
His paintings are so detailed you can clearly see the stitching.
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4. The rules of blackwork embroidery depend
(somewhat) on who you ask.
The most common materials traditionally are black thread and
light-colored linen.
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5. The stitching is traditionally done with “double running
stitch”.
1 2 3 4 5 6 7 8
Also sometimes back stitch, but we will be talking about double
running stitch.
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6. The stitching is traditionally done with “double running
stitch”.
1 2 3 4 5 6 7 8
15 14 13 12 11 10 9
Also sometimes back stitch, but we will be talking about double
running stitch.
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7. The stitching is traditionally done with “double running
stitch”.
5 6 15 16 25 26
4 14 24
1 2 3 11 12 13 21 22 23 31 32
39 38 37 7 36 35 17 34 33 27
8 18 28
9 10 19 20 29 30
Also sometimes back stitch, but we will be talking about double
running stitch.
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8. 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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9. 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”.
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10. An (undirected) graph may be associated to a digraph
by forgetting about the ordering of the pairs.
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11. A (symmetric) digraph may be associated to a graph
by including both possible directions of each edge.
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12. A walk on a graph is a finite alternating sequence of
vertices and edges x0 , {x0 , x1 }, x1 , . . . , {xn−1 , xn }, xn .
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13. A graph is connected if there is a walk between any
two vertices.
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14. A directed walk on a digraph is a finite sequence of
vertices and edges x0 , (x0 , x1 ), x1 , . . . , (xn−1 , xn ), xn .
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15. A Eulerian circuit on a digraph is a directed walk with
every edge used exactly once and x0 = xn .
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16. Every symmetric digraph associated to a connected
graph is Eulerian, i.e., has an Eulerian circuit.
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17. Having an Eulerian digraph is necessary for a pattern
to be reversibly stitchable, but is it sufficient?
The two different directions have to lie on opposite sides of the fabric.
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18. Having an Eulerian digraph is necessary for a pattern
to be reversibly stitchable, but is it sufficient?
5 6 15 16 25 26
4 14 24
1 2 3 11 12 13 21 22 23 31 32
39 38 37 7 36 35 17 34 33 27
8 18 28
9 10 19 20 29 30
The two different directions have to lie on opposite sides of the fabric.
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19. 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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20. A Holbeinian circuit on a digraph is an Eulerian circuit
where all (x, y ) and (y , x) have opposite parities.
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21. Theorem: Every symmetric digraph associated to a
connected graph is Holbeinian.
This will not be a surprise to any embroiderers.
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22. Theorem: Every symmetric digraph associated to a
connected graph is Holbeinian.
There are several ways we could prove this theorem.
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23. One way is to give an algorithm which produces the
circuit. 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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24. One way is to give an algorithm which produces the
circuit. 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 not
been traversed (and such that (z, y ) either has not been
traversed or was traversed with parity opposite the current
parity) 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 with
parity opposite the current parity).
c. If there are no allowed moves as above, terminate the
algorithm.
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25. Theorem: Tarry’s Algorithm with parity always
produces a Holbeinian circuit.
Example
3 4
17 16
19 18 6 5 15
1 2 12 9 7 8
11 10
13 14
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26. (False) Conjecture: Every Holbeinian circuit can be
produced from Tarry’s algorithm.
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27. (False) Conjecture: Every Holbeinian circuit can be
produced from Tarry’s algorithm.
Counterexample
3 4
5 12
1 2 6 11 15 16
19 18 10 17 13
9 14
7 8
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28. Fact: It is not possible for a “local” algorithm to tell
whether it is producing an Eulerian circuit or not.
1 2
5 9 4
6 3
8 7
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29. Fact: It is not possible for a “local” algorithm to tell
whether it is producing an Eulerian circuit or not.
1 2 1 2
4
5 9 4
6 3 3
8 7
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30. Conjecture: There is a “local” algorithm with an
“oracle” which produces every Holbeinian circuit.
What sort of oracle?
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31. What if you would like to stitch a pattern so that the top
and bottom threads go in the same direction?
1
15
8
4 3 7 2
11 14 10 9
12 13
5 6
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32. A multigraph of multiplicity 2, or 2-multigraph, is an
(undirected) multigraph where each edge appears
exactly twice.
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33. An Aragonian circuit of a 2-multigraph is an Eulerian
circuit where the two edges {x, y } are traversed in the
same order but with opposite parities.
(Same direction, opposite sides of the fabric.)
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34. Theorem: A multigraph of multiplicity 2 is Aragonian if
and only if the associated graph is Eulerian and has a
circuit of odd length.
Example
1
15
8
4 3 7 2
11 14 10 9
12 13
5 6
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35. Theorem: A multigraph of multiplicity 2 is Aragonian if
and only if the associated graph is Eulerian and has a
circuit of odd length.
Non-example
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36. What if some of the edges need to be stitched in a
“Holbeinian way” and some in an “Aragonian way”?
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37. There are lots of other questions to be asked!
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38. Thanks, and happy stitching!
[1] Joshua Holden, The Graph Theory of Blackwork Embroidery, Making
Mathematics with Needlework (sara-marie belcastro and Carolyn
Yackel, 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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