Supplementary material for the paper Genus of the Cartesian product of triangles by M. Kotrbčík and T. Pisanski.


We denote the Cartesian product of n triangles by Gn

G2:
We provide selected embeddings of G2 related to Theorem 3.8, Table 5, and Table 6. In particular:


G3:
We provide all embeddings related to Theorem 3.9 and Table 7. In particular:
G4:
We provide all embeddings related to Theorem 3.7, Table 3, Table 4. In particular:


Formats used.

Rotation schemes:
We use the following format for all rotation schemes with the single exception of the embedding from Table 4.

Each embedding is given by n+2 lines, where n is the number of vertices of the graph. The lines are as follows.

1st line: a single number n, the number of vertices of the graph;
2nd line: the string "genus: X number of faces: Y" where X is the genus of the embedding and Y is the number of faces of the embedding;

followed by n lines giving the rotations. Each of these lines has the form
i-th line: the degree of the i-th vertex followed by the rotation at the vertex. The rotation is given as a space-separated permutation of neighbours of the vertex.

For example, if the i-th line contains the string "4 0 1 3 2" it means that the vertex i has degree 4, the neighbours of the i-th vertex are 0, 1, 2, 3, and that the rotation at the i-th vertex is (0,1,3,2).

The vertices are numbered from zero.


The exceptional embedding of G4, given in Table 4, uses the following format. The graph G4 has 81 vertices and there are 81 lines, one for each vertex. The i-th line has the form

ci: n1, n2, n3, n4, n5, n6, n7, n8

where ci is the word corresponding to the coordinates of i-th vertex (see below) and n1, ..., n8 are the words corresponding to the neighbours of the i-th vertex and the rotation at i is (n1, ..., n8). (The words in the rotation are separated by a colon followed by a whitespace.)

We represent the coordinates of vertices as words over {0,1,2} as follows. The representation of i-th vertex is the reverse of fixed-length base-3 representation of the number i, where the length is n if the graph is Gn.

For example, in G4 for the vertex number 7 we have 7 = 0021, so the vertex 7 is represented by the string 1200. For convenience, the translation tables for n = 2, 3, and 4 are provided.


Face distributions:


Each face distribution is given by two lines, each containing precisely 27 integers.

The i-th integer in the first line is the number of faces of length i+2. (All our graphs are simple and thus cannot contain faces of length 1 and 2. Therefore, the numbers of faces with length 1 and 2 are ommited).

The i-th integer in the second line is the number of repetitive faces among the faces of length i+2. A face is repetitive if it contains any vertex more than once.