This page contains some catalogues of directed graphs prepared by Brendan McKay.

Here are the digraphs up to 6 vertices with loops either not allowed or allowed. The format is digraph6.

**Loops not allowed**

1 vertex (1)

2 vertices (3)

3 vertices (16)

4 vertices (218)

5 vertices (9608)

6 vertices (1540944)

**Loops allowed**

1 vertex (2)

2 vertices (10)

3 vertices (104)

4 vertices (3044)

5 vertices (291968)

6 vertices (96928992, gzipped)

By an oriented graph we mean a digraph with no cycles of length 2. Here are the oriented graphs up to 7 vertices with loops not allowed and 6 vertices with loops allowed. The format is digraph6.

**Loops not allowed**

1 vertex (1)

2 vertices (2)

3 vertices (7)

4 vertices (42)

5 vertices (582)

6 vertices (21480)

7 vertices (2142288, gzipped)

**Loops allowed**

1 vertex (2)

2 vertices (7)

3 vertices (44)

4 vertices (558)

5 vertices (16926)

6 vertices (1319358)

Here are the non-isomorphic tournaments up to 10 vertices. Each is given as the upper triangle of the adjacency matrix in row order, on one line without spaces.

2 vertices (1)

3 vertices (2)

4 vertices (4)

5 vertices (12)

6 vertices (56)

7 vertices (456)

8 vertices (6880)

9 vertices (191536)

10 vertices (gzipped) (9733056)

Here are the non-isomorphic self-converse tournaments (self-complementary oriented graphs) up to 13 vertices. Each is given as the upper triangle of the adjacency matrix in row order, on one line without spaces.

2 vertices (1)

3 vertices (2)

4 vertices (2)

5 vertices (8)

6 vertices (12)

7 vertices (88)

8 vertices (176)

9 vertices (2752)

10 vertices (8784)

11 vertices (279968)

12 vertices (gzipped) (1492288)

13 vertices (gzipped) (95458560)

A tournament of odd order n is **regular** if the out-degree of each
vertex is (n-1)/2. A tournament of even order n is **semi-regular**
if the out-degree of each vertex is n/2-1 or n/2. Here are the regular
and semi-regular tournaments of order up to 13.

3 vertices (1)

4 vertices (1)

5 vertices (1)

6 vertices (5)

7 vertices (3)

8 vertices (85)

9 vertices (15)

10 vertices (13333)

11 vertices (1223)

12 vertices (gzipped) (19434757):
Part a
Part b (these unpack to 1.3GB altogether)

13 vertices (gzipped) (1495297)

A regular tournament is **doubly-regular** if each pair of vertices is
jointly connected to exactly (n-3)/4 others. The order must be one less
than a multiple of 4. These tournaments are related to skew Hadamard
matrices. Ted Spence was the first to find them up to 27 vertices
but I have recomputed them. The incomplete lists on larger sizes were
computed using skew Hadamard matrices from
Christos Koukouvinos's catalogue.

3 vertices (1)

7 vertices (1)

11 vertices (1)

15 vertices (2)

19 vertices (2)

23 vertices (37)

27 vertices (722)

31 vertices (5 incomplete)

35 vertices (486 incomplete)

39 vertices (1560 incomplete)

43 vertices (2178 incomplete)

47 vertices (3 incomplete)

51 vertices (gzipped) (36350 incomplete)

A tournament is **locally-transitive** if, for each vertex v,
the in-neighbourhood and the out-neighbourhood of v are both
transitive tournaments.

Here are the non-isomorphic locally-transitive tournaments up to 20 vertices. Each is given as the upper triangle of the adjacency matrix in row order, on one line without spaces.

3 vertices (2)

4 vertices (2)

5 vertices (4)

6 vertices (6)

7 vertices (10)

8 vertices (16)

9 vertices (30)

10 vertices (52)

11 vertices (94)

12 vertices (172)

13 vertices (316)

14 vertices (586)

15 vertices (1096)

16 vertices (gzipped) (2048)

17 vertices (gzipped) (3856)

18 vertices (gzipped) (7286)

19 vertices (gzipped) (13798)

20 vertices (gzipped) (26216)

Here are the acyclic digraphs up to 8 vertices. Each is given as the upper triangle of the adjacency matrix in row order, on one line without spaces. The lower triangle is zero; that is, the points are in topological order.

2 vertices (2)

3 vertices (6)

4 vertices (31)

5 vertices (302)

6 vertices (5984)

7 vertices (gzipped) (243668)

8 vertices (gzipped) part 1
part 2
part 3
part 4 (20286025)

Here are the connected Eulerian digraphs up to 8 vertices. There are no loops. The format is digraph6.

2 vertices (1)

3 vertices (3)

4 vertices (12)

5 vertices (90)

6 vertices (2162)

7 vertices (179098)

8 vertices (51110788, gzipped)

Here are the connected Eulerian oriented graphs up to 9 vertices. There are no loops. These differ from the previous list in that 2-cycles are forbidden. The format is digraph6.

3 vertices (1)

4 vertices (1)

5 vertices (4)

6 vertices (16)

7 vertices (175)

8 vertices (5274)

9 vertices (434017)

Here are the Hasse diagrams of posets up to 10 points.
There is one poset per line in a format like
"`10 4 56 57 78 79`". This means there are 10 points
0..9, and 4 covering relations that are 5<6, 5<7,
7<8 and 7<9. The full poset is the reflexive transitive
closure of the Hasse diagram. A program is available that
can make larger sizes, but the numbers
grow very quickly.

1 point (1)

2 points (2)

3 points (5)

4 points (16)

5 points (63)

6 points (318)

7 points (2045)

8 points (16999)

9 points (183231, gzipped)

10 points (2567284, gzipped)

See the Ramsey page.

Here are all the tournaments up to 49 vertices whose automorphism groups act transitively on the vertex set. Since all automorphisms of a tournament have odd order, the tournament must also have odd order. First we will give the circulant tournaments, which are those having (0 1 ... n-1) as an automorphism. Then we will give the vertex-transitive tournaments up to 49 vertices that are not circulant.

Thanks to Gordon Royle for helping with the permutation groups.

1 vertex (1 tournament)

3 vertices (1 tournament)

5 vertices (1 tournament)

7 vertices (2 tournaments)

9 vertices (3 tournaments)

11 vertices (4 tournaments)

13 vertices (6 tournaments)

15 vertices (16 tournaments)

17 vertices (16 tournaments)

19 vertices (30 tournaments)

21 vertices (88 tournaments)

23 vertices (94 tournaments)

25 vertices (205 tournaments)

27 vertices (457 tournaments)

29 vertices (586 tournaments)

31 vertices (1096 tournaments)

33 vertices (3280 tournaments)

35 vertices (5472 tournaments)

37 vertices (7286 tournaments)

39 vertices (21856 tournaments)

41 vertices (26216 tournaments)

43 vertices (49940 tournaments, gzipped)

45 vertices (174848 tournaments, gzipped)

47 vertices (182362 tournaments, gzipped)

49 vertices (399472 tournaments, gzipped)

21 vertices (22 tournaments)

25 vertices (9 tournaments)

27 vertices (237 tournaments)

39 vertices (3350 tournaments)

45 vertices (21776 tournaments)

49 vertices (8384 tournaments)

The **deck** of a digraph *G* is the multiset of its
vertex-deleted subdigraphs *G-v*. The elements of the deck
are specified only up to isomorphism. We say that *G* is
reconstructable (also, reconstructible) if *G* is uniquely
determined by its deck. Here we give all the known simple digraphs
which are not reconstructable, starting at 3 vertices.

3 vertices 2 pairs, 2 triples

4 vertices 8 pairs

5 vertices 9 pairs

6 vertices 9 pairs

On 7 vertices, all digraphs are reconstructable.

8 vertices 5 pairs

Those files are complete.
From 9 vertices onwards, all known nonreconstructable digraphs lie
in one of the infinite families found by Paul Stockmeyer. There
are 4 pairs for each order which is a power of 2, and 6 pairs
for each order which is the sum of two distinct powers of 2.
stock.c is a program in standard C
that writes the digraphs. Run it with one argument which is the
number of vertices.

The **reduced deck** of a digraph *G* is the same as the deck
except that isomorphs have been removed. Thus, the reduced deck
tells you which graphs appear in the deck up to isomorphism but
not how many times they appear. When only the reduced deck is available,
additional digraphs are not uniquely determined. The following
collection is complete up to 8 vertices.

3 vertices two sets of size 4

4 vertices one triple

5 vertices one pair

7 vertices 3 pairs

Page Master: Brendan McKay, brendan.mckay@anu.edu.au and https://users.cecs.anu.edu.au/~bdm.